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aught er   of  William 
tuart   SmithtU»S.Navy 


ENGINEERING  LIBRARY 


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THEORY  AND  CALCULATION 


ALTERNATING  CURRENT 
PHENOMENA  . 


BY 

CHARLES    PROTEUS    STEINMETZ 


WITH    THE    ASSISTANCE   OF 


ERNST   J.    BERG 


NEW    YORK 
THE   W.    J.    JOHNSTON    COMPANY 

253    BROADWAY 
l897 


COPYRIGHT,  1897, 

BY 
TUB  W.  J.  JOHNSTON  COMPANY. 


•V-a 
ENGINEERING  LIBRARY 


u 


TYPOGRAPHY    BY   C.   J.     PETERS   &   SON,    BOSTON. 


DEDICATED 

TO    THE 

MEMORY    OF    MY    FATHER, 
CARL    HEINRICH    STEINMETZ. 


PREFACE. 


THE  following  volume  is  intended  as  an  exposition  of 
the  methods  which  I  have  found  useful  in  the  theoretical 
investigation  and  calculation  of  the  manifold  phenomena 
taking  place  in  alternating-current  circuits,  and  of  their 
application  to  alternating-current  apparatus. 

While  the  book  is  not  intended  as  first  instruction  for 
a  beginner,  but  presupposes  some  knowledge  of  electrical 
engineering,  I  have  endeavored  to  make  it  as  elementary 
as  possible,  and  have  therefore  only  used  common  algebra 
and  trigonometry,  practically  excluding  calculus,  except  in 
§§  106  to  115  and  Appendix  II. ;  and  even  §§  106  to  115 
have  been,  paralleled  by  the  elementary  approximation  of 
the  same  phenomenon  in  §§  102  to  105. 

All  the  methods  used  in  the  book  have  been  introduced 
and  explicitly  discussed,  with  instances  of  their  application, 
the  first  part  of  the  book  being  devoted  to  this.  In  the  in- 
vestigation of  alternating-current  phenomena  and  apparatus, 
one  method  only  has  usually  been  employed,  though  the 
other  available  methods  are  sufficiently  explained  to  show 
their  application. 

A  considerable  part  of  the  book  is  necessarily  devoted 
to  the  application  of  complex  imaginary  quantities,  as  the 
method  which  I  found  most  useful  in  dealing  with  alternat- 
ing-current phenomena ;  and  in  this  regard  the  book  may  be 
considered  as  an  expansion  and  extension  of  my  paper  on 
the  application  of  complex  imaginary  quantities  to  electri- 
cal engineering,  read  before  the  International  Electrical  Con- 


VI  PREFACE. 

gress  at  Chicago,  1893.  The  complex  imaginary  quantity 
is  gradually  introduced,  with  full  explanations,  the  algebraic 
operations  with  complex  quantities  being  discussed  in  Ap- 
pendix L,  so  as  not  to  require  from  the  reader  any  previous 
knowledge  of  the  algebra  of  the  complex  imaginary  plane. 

While  those  phenomena  which  are  characteristic  to  poly- 
phase systems,  as  the  resultant  action  of  the  phases,  the 
effects  of  unbalancing,  the  transformation  of  polyphase  sys- 
tems, etc.,  have  been  discussed  separately  in  the  last  chap- 
ters, many  of  the  investigations  in  the  previous  parts  of  the 
book  apply  to  polyphase  systems  as  well  as  single-phase 
circuits,  as  the  chapters  on  induction  motors,  generators, 
synchronous  motors,  etc. 

A  part  of  the  book  is  original  investigation,  either  pub- 
lished here  for  the  first  time,  or  collected  from  previous 
publications  and  more  fully  explained.  Other  parts  have 
been  published  before  by  other  investigators,  either  in  the 
same,  or  more  frequently  in  a  different  form. 

I  have,  however,  omitted  altogether  literary  references, 
for  the  reason  that  incomplete  references  would  be  worse 
than  none,  while  complete  references  would  entail  the  ex- 
penditure of  much  more  time  than  is  at  my  disposal,  with- 
out offering  sufficient  compensation  ;  since  I  believe  that 
the  reader  who  wants  information  on  some  phenomenon  or 
apparatus  is  more  interested  in  the  information  than  in 
knowing  who  first  investigated  the  phenomenon. 

Special  attention  has  been  given  to  supply  a  complete 
and  extensive  index  for  easy  reference,  and  to  render  the 
book  as  free  from  errors  as  possible.  Nevertheless,  it  prob- 
ably contains  some  errors,  typographical  and  otherwise ; 
and  I  will  be  obliged  to  any  reader  who  on  discovering  an 
error  or  an  apparent  error  will  notify  me. 

I  take  pleasure  here  in  expressing  my  thanks  to  Messrs. 
W.  D.  WEAVER,  A.  E.  KENNELLY,  and  TOWNSEND  WOL- 
COTT,  for  the  interest  they  have  taken  in  the  book  while  in 
the  course  of  publication,  as  well  as  for  the  valuable  assist- 


PREFACE.  Vll 

ance  given  by  them  in  correcting  and  standardizing  the  no- 
tation to  conform  with  the  international  system,  and  numer- 
ous valuable  suggestions  regarding  desirable  improvements. 
Thanks  are  due  also  to  the  publishers,  who  have  spared 
no  effort  or  expense  to  rniake  the  book  as  creditable  as  pos- 
sible mechanically. 

CHARLES   PROTEUS   STEINMETZ. 

January,  1897. 


/CONTENTS. 


CHAP.  I.    Introduction.—  ,    ,  ,-•  j, 

§    1,  p.     1.     Fundamental  laws  of  continuous  current  circuits, 
§    2,  p.     2.     Impedance,  reactance,  effective  resistance. 
§    3,  p.     3.     Electro-magnetism  as  source  of  reactance., 
§    4,  p.     5.     Capacity  as  source  of  reactance. 

§    5,   p.     6.     Joule's  law  and  power  equation  of  alternating  circuit. 
§    6,  p.     6.     Fundamental    wave    and     higher     harmonics,    alternating 

waves  with  and  without  even  harmonics. 
§    7,  p.     9.     Alternating  waves  as  sine  waves. 

CHAP.  II.    Instantaneous  Values  and  Integral  Values. — 

§    8,  p.   11.     Integral  values  of  wave. 

§    9,  p.   13.     Ratio  of  mean  to  maximum  to  effective  value  of  wave. 

CHAP.  III.  Law  of  Electro-magnetic  Induction. — 

§'11,  p.  16.      Induced  E.M.F.  mean  value. 

§  12,  p.  17.     Induced  E.M.F.   effective  value. 

§  13,  p.  18.     Inductance  and  reactance. 

CHAP.  IV.    Graphic  Representation.  — 

§  14,  p.   19.     Polar  characteristic  of  alternating  wave. 

§  15,  p.  20.     Polar  characteristic  of  sine  wave. 

§  16,  p.  21.     Parallelogram  of  sine  waves,  Kirchhoff's  laws,  and  energy 

equation. 

§  17,  p.  23.     Non-inductive  circuit  fed  over  inductive  line,  instance. 
§  18,  p.  24.     Counter  E.M.F.   and  component  of  impressed  E.M.F. 
§  19,  p.  26.     Continued. 
§20,  p.   26.     Inductive  circuit  and  circuit  with  leading  current  fed  over 

inductive  line.     Alternating-current  generator. 

§  21,  p.   28.      Polar  diagram  of  alternating-current  transformer,  instance. 
§22,  p.  30.      Continued. 

CHAP.  V.    Symbolic  Method. — 

§  23,  p.  33.      Disadvantage  of  graphic  method  for  numerical  calculation. 

§  24,  p.   34.     Trigonometric  calculation. 

§  25,   p.   34.      Rectangular  components  of  vectors. 

§  26,   p.   36.      Introduction  of  j  as  distinguishing  index. 

§27,  p.   36.     Rotation  of  vector  by  180°  and  90°.    j  =  V^l. 


X  CONTENTS. 

CHAP.  V.     Symbolic  Method  —  Continued.  — 

§  28,  p.  37.     Combination  of  sine  waves  in  symbolic  expression. 

§  29,  p.  38.     Resistance,  reactance,  impedance  in  symbolic  expression. 

§  30,  p.  40.     Capacity  reactance  in  symbolic  representation. 

§  31,  p.  40.     Kirchhoff's  laws  in  symbolic  representation. 

§  32,  p.  41.     Circuit  supplied  over  inductive  line,  instance. 

CHAP.  VI.    Topographic  Method.— 
§  33,  p.  43.     Ambiguity  of  vectors. 
§  34,  p.  44.     Instance  of  a  three-phase  system. 
§  35,  p.  46.     Three-phase  generator  on  balanced  load. 
§  36,  p.  48.     Three-phase  generator  on  unbalanced  load. 
§  37,  p.  50.     Quarter-phase  system  with  common  return. 

CHAP.  VII.    Admittance,  Conductance,  Susceptance.  — 

§  38,  p.  52.  Combination  of  resistances  and  conductances  in  series  and 
in  parallel. 

§  39,  p.  53.  Combination  of  impedances.  Admittance,  conductance, 
susceptance. 

§  40,  p.  54.  Relation  between  impedance,  resistance,  reactance,  and 
admittance,  conductance,  susceptance. 

§  41,  p.  56.  Dependence  of  admittance,  conductance,  susceptance,  upon 
resistance  and  reactance.  Combination  of  impedances  and  ad- 
mittances. 

CHAP.  VIII.     Circuits  containing  Resistance,  Inductance,  and  Ca- 
pacity. — 

§42,  p.  58.     Introduction. 
§  43,  p.  58.     Resistance  in  series  with  circuit. 
§  44,  p.  60.     Discussion  of  instances. 
§45,  p.  61.     Reactance  in  series  with  circuit. 
§  46,   p.   64.     Discussion  of  instances. 
§  47,  p.   66.      Reactance  in  series  with  circuit. 
§  48,  p.   68.     Impedance  in  series  with  circuit. 
§49,  p.  69.     Continued. 
§  50,  p.   71.     Instance. 

§  51,  p.   72.     Compensation  for  lagging  currents  by  shunted  condensance. 
§  52,  p.  73.     Complete  balance  by  variation  of  shunted  condensance. 
§  53,  p.   75.     Partial  balance  by  constant  shunted  condensance. 
§54,  p.   76.      Constant  potential  —  constant  current  transformation. 
§55,   p.   79.     Constant  current — constant  potential  transformation. 
§56,  p.  81.     Efficiency   of    constant    potential — constant  current  trans- 
formation. 

CHAP.  IX.  Resistance  and  Reactance  of  Transmission  Lines.  — 

§57,  p.  83.     Introduction. 

§58,  p.  84.     Non-inductive  receiver  circuit  supplied  over  inductive  line. 

§59,  p.  86.     Instance. 


CONTENTS.  XI 

CHAP.  IX.    Resistance    and    Reactance    of   Transmission   Lines  — 

Continued.  — 

§  60,  p.     87.     Maximum  power  supplied  over  inductive  line. 

§  61,  p.  88.  Dependence  of  output  upon  the  susceptance  of  the  re- 
ceiver circuit. 

§  62,  p.  89.  Dependence  oi<  output  upon  the  conductance  of  the  re- 
ceiver circuit. 

§  63,  p.     90.     Summary. 

§  64,  p.     92.     Instance. 

§  65,  p.     93.     Condition  of  maximum  efficiency. 

§  66,  p.     96.     Control  of  receiver  voltage  by  shunted  susceptance. 

§  67,  p.     97.     Compensation  for  line  drop  by  shunted  susceptance. 

§  68,  p.     97.     Maximum  output  and  discussion. 

§  69,  p.     98.     Instances. 

§70,  p.  101.     Maximum  rise  of  potential  in  receiver  circuit. 

§  71,  p.   102.     Summary  and  instances. 

CHAP.  X.    Effective  Resistance  and  Reactance. — 

§  72,  p.  104.  Effective  resistance,  reactance,  conductance,  and  suscep- 
tance. 

§  73,  p.   105.     Sources  of  energy-losses  in  alternating-current  circuits. 

§74,  p.   106.     Magnetic  hysteresis. 

§  75,   p.   107.     Hysteretic  cycles  and  corresponding  current  waves. 

§76,  p.  111.  Action  of  air-gap  and  of  induced  current  on  hysteretic 
distortion. 

§77,  p.  111.     Equivalent  sine  wave  and  wattless  higher  harmonic. 

§  78,  p.   113.     True  and  apparent  magnetic  characteristic. 

§  79,  p.   115.     Angle  of  hysteretic  advance  of  phase. 

§  80,   p.   116.     Loss  of  energy  by  molecular  magnetic  friction. 

§  81,  p.   119.      Effective  conductance,  due  to  magnetic  hysteresis. 

§  82,  p.  122.  Absolute  admittance  of  ironclad  circuits  and  angle  of 
hysteretic  advance. 

§  83,   p.   124.     Magnetic  circuit  containing  air-gap. 

§  84,  p.   125.     Electric  constants  of  circuit  containing  iron. 

§  85,  p.   127.     Conclusion. 

CHAP.  XI.    Foucault  or  Eddy  Currents.  — 

§  86,  p.   129.     Effective  conductance  of  eddy  currents. 

§  87,  p.   130.     Advance  angle  of  eddy  currents. 

§  88,  p.   131.     Loss  of  power  by  eddy  currents,  and  coefficient  of  eddy 

currents. 

§  89,   p.   131.     Laminated  iron. 
§  90,   p.   133.      Iron  wire. 

§  91,  p.    135.     Comparison  of  sheet  iron  and  iron  wire. 
§  92,  p.   136.     Demagnetizing  or  screening  effect  of  eddy  currents. 
§93,   p.   138.     Continued. 
§  94,   p.    138.      Large  eddy  currents. 


Xll  CONTENTS. 

CHAP.  XI.    Foucault  or  Eddy  Currents  —  Continued. — 

§    95,  p.  139.     Eddy  currents    in  conductor    and   unequal  current    dis- 
tribution. 

§    96,  p.  140.     Continued. 
§    97,  p.  142.     Mutual  inductance. 

§    98,  p.   144.     Dielectric  and  electrostatic  phenomena. 
§    99,  p.  145.     Dielectric   hysteretic  admittance,   impedance,   lag,  etc. 
§  100,  p.   147.     Electrostatic  induction  or  influence. 
§  101,  p.   148.     Energy  components  and  wattless  components. 

CHAP.  XII.    Distributed    Capacity,    Inductance,    Resistance,    and 
Leakage.  — 

§  102,  p.   150.     Introduction. 

§  103,  p.   151.     Magnitude  of  charging  current  of  transmission  lines. 

§  104,  p.   152.     Line    capacity    represented   by    one    condenser    shunted 
'across  middle  of  line. 

§  105,  p.   153.     Line  capacity  represented  by  three  condensers. 

§  106,  p.   155.     Complete    investigation    of    distributed   capacity,    induc- 
tance, leakage,  and  resistance. 

§  107,  p.   157.     Continued. 

§  108,  p.  158.     Continued. 

§  109,  p.   158.     Continued. 

§  110,  p.   159.     Continued. 

§  111,  p.   161.     Continued. 

§  112,  p.   161.     Continued. 

§  113,  p.   162.     Difference  of  phase  at  any  point  of  line. 

§  114,  p.   163.     Instance. 

§  115,  p.   165.     Particular    cases,    open    circuit    at    end    of    line,    line 
grounded    at    end,    infinitely    long    conductor,    generator     feeding 
into  closed  circuit. 
CHAP.  XIII.    The  Alternating-Current  Transformer.— 

§  116,  p.  167.  General. 

§  117,  p.  167.  Mutual  inductance  and  self-inductance  of  transformer. 

§  118,  p.  168.  Magnetic  circuit  of  transformer. 

§  119,  p.   169.  Continued. 

§  120,  p.   170.  Polar  diagram  of  transformer. 

§  121,  p.   172.  Instance. 

§  122,  p.   176.  Diagram  for  varying  load. 

§  123,  p.   177.  Instance. 

§  124,  p.   178.  Symbolic  method,   equations. 

§  125,  p.   180.  Continued. 

§  126,  p.    182.  Apparent     impedance     of     transformer.        Transformer 

equivalent  to  divided  circuit. 

§  127,  p.   183.  Continued. 

§  128,  p.   186.  Transformer  on  non-inductive  load. 

§  129,  p.   188.  Constants  of  transformer  on  non-inductive  load. 

§  130,  p.   191.  Numerical  instance. 


CONTENTS.  xili 

CHAP.  XIV.    General  Alternating-Current  Transformer.  — 

§  131,  p.   194.  Introduction. 

§  132,  p.  194.  Magnetic  cross-flux  or  self-induction  of  transformer. 

§  133,  p.   195.  Mutual  flux  of  transformer. 

§  134,  p.   195.  Difference  of   frequency  between  primary  and   second- 
ary of  general  alternate*fcurrent  transformer. 

§  135,  p.   195.  Equations  of  general  alternate-current  transformer. 

§  136,  p.  201.  Power,  output  and  input,  mechanical  and  electrical. 

§  137,  p.  202.  Continued. 

§  138,  p.  203.  Speed  and  output. 

§  139,  p.  205.  Numerical  instance. 

CHAP.  XV.     Induction  Motor.  — 

§  140,  p.  207.  Slip  and  secondary  frequency. 

§  141,  p.  208.  Equations  of  induction  motor. 

§  142,  p.  209.  Magnetic  flux,  admittance,  and  impedance. 

§143,  p.  211.  E.M.F. 

§  144,  p.  213.  Graphic  representation. 

§  145,  p.  214.  Continued. 

§  146,  p.  216.  Torque  and  power. 

§  147,  p.  218.  Power  of  induction  motors. 

§  148,  p.  219.  Maximum  torque. 

§  149,  p.  221.  Continued. 

§  150,  p.  222.  Maximum  power. 

§  151,  p.   224.  Starting  torque. 

§  152,  p.    225.*  Continued. 

§  153,  p.   227.  Starting  resistance. 

§  154,   p.  228.  Synchronism. 

§  155,  p.  228.  Near  synchronism. 

§  156,  p.  229.  Induction  generator. 

§  157,  p.  229.  Comparison    of    induction    generator    and    synchronous 

generator. 

§  158,   p.  230.  Numerical  instance  of  induction  motor. 

CHAP.  XVI.    Alternate-Current  Generator.— 

§  159,  p.  234.  General. 

§  1'60,  p.   235.  Magnetic  reaction  of  lag  and  lead. 

§  161,  p.  237.  Self-inductance  of  alternator. 

§  162,  p.  238.  Synchronous  reactance. 

§  163,  p.  238.  Equations  of  alternator. 

§  164,  p.   239.  Numerical   instance,  field  characteristic. 

§  165,  p.   244.  Dependence  of  terminal  voltage  on  phase  relation. 

§  166,  p.  244.  Constant  potential  regulation. 

§  167,  p.   246.  Constant  current   regulation,  maximum  output. 

CHAP.  XVII.     Synchronizing  Alternators.— 

§  168,   p.  248.  Introduction. 

§  169,  p.  248.  Rigid  mechanical  connection. 


XIV 


CONTENTS. 


CHAP.  XVII.     Synchronous  Motor  —  Continued.  — 
§  170,  p.  248.     Uniformity  of  speed. 
§  171,  p.  249. 
§  172,  p.  250. 


Synchronizing. 

Running  in  synchronism. 


§  173,  p.  250.     Series  operation  of  alternators. 

§  174,  p.  251.     Equations  of  synchronous  running  alternators,  synchro- 

nizing power. 

§  175,  p.   254.     Special  case  of  equal  alternators  at  equal  excitation. 
§  176,  p.  257.     Numerical  instance. 

CHAP.  XVIII.    Synchronous  Motor.— 

§  177,  p.  258.     Graphic  method. 


178,  p. 

179,  p. 

180,  p. 

181,  p. 

182,  p. 

183,  p. 

184,  p. 


260.  Continued. 

262.  Instance. 

263.  -Constant  impressed  E.M.F.  and  constant  current. 
266.  Constant  impressed  and  counter  E.M.F. 

269.  Constant  impressed  E.M.F.  and  maximum  efficiency. 

271.  Constant  impressed  E.M.F.  and  constant  output. 

275.  Analytical  method.     Fundamental  equations  and  power 
characteristic. 

185,  p.  279.  Maximum  output. 

186,  p.  280.  No  load. 

282.  Minimum  current. 

284.  Maximum  displacement  of  phase. 

286.  Constant  counter  E.M.F. 

286.  Numerical  instance. 

288.  Discussion  of   results. 


187, 
188, 
189, 
190, 


§  191,  p. 

CHAP.  XIX. 
§  192,  p. 
§193,  p. 
§  194,  p. 
§  195,  p. 
§  196,  p. 

§  197,  P: 

§  198,  p. 
§  199,  p. 
§200,  p. 
§201,  p. 
§202,  p. 
§203,  p. 

CHAP.  XX. 
§204,  p. 


§205, 
§206, 
§207, 


Commutator  Motors.— 

291.  Types  of  commutator  motors. 

291.  Repulsion  motor  as  induction  motor. 

293.  Two  types  of  repulsion  mot9rs. 

295.  Definition  of  repulsion  motor. 

296.  Equations  of  repulsion  motor. 

297.  Continued. 

298.  Power  of  repulsion  motor.     Instance. 
300.  Series  motor,  shunt  motor. 

303.  Equations  of  series  motor. 

304.  Numerical  instance. 

305.  Shunt  motor. 

307.  Power  factor  of  series  motor. 

Reaction  Machines.  — 

308.  General  discussion. 

309.  Energy  component  of  reactance. 


p.  309.*    Hysteretic  energy  component  of  reactance, 
p.  310.     Periodic  variation  of  reactance. 


CONTENTS.  XV 

CHAP.  XX.    Reaction  Machines  —  Continued.  — 
§  208,  p.  312.     Distortion  of  wave-shape. 
§  209,  p.  314.     Unsymmetrical  distortion  of  wave-shape. 
§  210,  p.  315.     Equations  of  reaction  machines. 
§  211,  p.  317.     Numerica>  instance. 

CHAP.  XXI.    Distortion  of  Wave-Shape  and  its  Causes. — 

§  212,  p.  320.  Equivalent  sine  wave. 

§  213,  p.  320.  Cause  of  distortion. 

§214,  p.  321.  Lack  of  uniformity  and  pulsation  of  magnetic  field. 

§215,  p.  324.  Continued. 

§  216,  p.  327.  Pulsation  of  reactance. 

§  217,  p.  327.  Pulsation  of  reactance  in  reaction  machine. 

§  218,  p.   329.  General  discussion. 

§  219,  p.  329.  Pulsation  of  resistance,  arc. 

§220,  p.  331.  Instance. 

§  221,  p.  332.  Distortion  of  wave-shape  by  arc. 

§222,  p.  333.  Discussion. 

CHAP.  XXII.    Effects  of  Higher  Harmonics. — 

§  223,  p.  334.  Distortion  of   wave-shape  by  triple  and  quintuple  har- 
monics. Some  characteristic  wave-shapes. 

§  224,  p.  337.  Effect  of  self-induction  and  capacity  on  higher  harmonics. 

§  225,  p.  338.  Resonance  due  to  higher  harmonics  in  transmission  lines. 

§226,  p.  341.  Power  of  complex  harmonic  waves. 

§227,  p.   341.  Three-phase  generator. 

§  228,  p.  343.  Decrease  of  hysteresis  by  distortion  of  wave-shape. 

§  229,  p.  343.  Increase  of  hysteresis  by  distortion  of  wave-shape. 

§230,  p.  344.  Eddy   currents. 

§  231,  p.  344.  Effect  of  distorted  waves  on  insulation. 

CHAP.  XXIII.     General  Polyphase  Systems.  - 

§  232,  p.   346.     Definition    of   systems,   symmetrical   and   unsymnvetrical 

systems. 
§  233,  p.  346.     Flow   of    power.      Balanced    and    unbalanced   systems. 

Independent  and  interlinked  systems.      Star  connection  and  ring 

connection. 
§  234,  p.   348.     Classification  of  polyphase  systems. 

CHAP.  XXIV.     Symmetrical  Polyphase  Systems.  — 

§  235,  p.   350.     General  equations  of  symmetrical  system. 
§  236,   p.  351.      Particular  syste'ms. 

§  237,  p.  352.     Resultant  M.M.F.  of  symmetrical  system. 
§  238,  p.  355.      Particular  systems. 

CHAP.  XXV.     Balanced  and  Unbalanced  Polyphase  Systems.  — 
§  239,   p.   356.      Flow   of  power  in  single-phase  system. 
§  240,   p.   357.     Flow  of  power  in  polyphase  systems,  balance  factor  of 

system. 


xvi  CONTENTS. 

CHAP.  XXV.    Balanced  and  Unbalanced  Polyphase  Systems  —  Con- 
tinued. — 

§241,  p.  358.     Balance  factor. 

§  242,  p.  358.     Three-phase  system,  q-uarter-phase  system. 
§  243,  p.  359.     Inverted  three-phase  system. 
§  244,  p.  360.     Diagrams  of  flow  of  power. 
§  245,  p.  363.     Monocycjic  and  polycyclic  systems. 
§  246,  p.   363.      Power  characteristic  of  alternating-current  system. 
§  247,  p.  364.     The  same  in  rectangular  coordinates. 
§  248,  p.   366.     Main  power  axes  of  alternating-current  system. 

CHAP.  XXVI.     Interlinked  Polyphase  Systems.  — 
§  249,  p.   368.      Interlinked  and  independent  systems. 
§  250,  p.  368.     Star    connection    and    ring    connection.       Y    connection 

and  delta  connection. 
§251,   p.  370.     Continued. 

§  252,  p.  371.     Star  potential  and  ring  potential.      Star  current  and  ring 
.  I  current.      Y   potential   and  Y  current,    delta   potential    and   delta 

current. 

§  253,   p.  371.     Equations  of  interlinked  polyphase  systems. 
§  254,  p.  373.     Continued. 

CHAP.  XXVII.    Transformation  of  Polyphase  Systems.  — 
§  255,  p.  376.     Constancy  of  balance  factor. 

§  256,  p.  376.     Equations  of  transformation  of  polyphase  systems. 
§  257,  p.  378.     Three-phase,  quarter-phase  transformation. 
§  258,  p.  379.     Transformation  with  change  of  balance  factor. 

CHAP.  XXVIII.    Copper  Efficiency  of  Systems.— 

§  259,  p.   380.     General  discussion. 

§  260,  p.  381.  Comparison  on  the  basis  of  equality  of  minimum  differ- 
ence of  potential. 

§  261,  p.  386.  Comparison  on  the  basis  of  equality  of  maximum  differ- 
ence of  potential. 

§262,  p.  388.     Continued. 

CHAP.  XXIX.    Three-phase  System.  — 

§  263,  p.  390.      General  equations. 

§  264,  p.  393.  Special  cases  :  balanced  system,  one  branch  loaded, 
two  branches  loaded. 

CHAP.  XXX.    Quarter-phase  System.  — 
§265,  p.  395.     General  equations. 
§  266,  p.  396.     Special  cases  :   balanced  system,  one  branch  loaded. 

APPENDIX  I.     Algebra  of  Complex  Imaginary  Quantities.  — 
§  267,  p.  401.     Introduction. 
§  268,   p.  401.     Numeration,   addition,   multiplication,  involution. 


CONTENTS.  xvii 

APPENDIX  I.     Algebra    of    Complex    Imaginary    Quantities  —  Con- 
tinued. 

§  269,   p.   402.  Subtraction,   negative  number. 

§  270,   p.   403.  Division,   fraction. 

§  271,   p.  403.  Evolution  and  logarithmation. 

§  272,  p.  404.  Imaginary  uni<^  complex  imaginary  number. 

§273,  p.   404.  Review. 

§  274,  p.  405.  Algebraic  operations  with  complex  quantities. 

§  275,  p.  406.  Continued. 

§  276,  p.  407.  Roots  of  the  unit. 

§  277,  p.  407.  Rotation. 

§  278,  p.  408.  Complex  imaginary  plane. 

APPENDIX  II.     Oscillating  Currents.  — 

§279,   p.   409.  Introduction. 

§  280,  p.  410.  General  equations. 

§  281,  p.  411.  Polar  coordinates. 

§  282,  p.  412.  Loxodromic  spiral. 

§  283,  p.  413.  Impedance  and  admittance. 

§284,  p.  414.  Inductance. 

§  285,  p.  414.  Capacity. 

§286,  p.  415.  Impedance. 

§287,  p.  416.  Admittance. 

§  288,  p.  417.  Conductance  and  susceptance. 

§  289,  p.  418.  Circuits  of  zero  impedance. 

§290,   p.  418.  Continued. 

§  291,  p.  419.  Origin  of  oscillating  currents. 

§292,  p.  420.  Oscillating  discharge. 

§  293,  p.  421.  Oscillating  discharge  of  condensers. 

§  294,   p.  422.  Oscillating  current  transformer. 

§  295,  p.  424.  Fundamental  equations  thereof. 


:.*'.  '  Jfr 

THEORY   AND   CALCULATION 

OF 

ALTERNATING-CURRENT   PHENOMENA. 


CHAPTER    I. 

INTRODUCTION. 

1.  IN  the  practical  applications  of  electrical  energy,  we 
meet  with  two  different  classes  of  phenomena,  due  respec- 
tively to  the  continuous  current  and  to  the  alternating 
current. 

The  continuous-current  phenomena  have  been  brought 
within  the  realm  of  exact  analytical  calculation  by  a  few 
fundamental  laws  :  — 

1.)  Ohm's  law  :  i  =  e  /  r,  where  r,  the  resistance,  is  a 
constant  of  the  circuit. 

2.)  Joule's  law:  P  =  i*r,  where  P  is  the  rate  at  which 
energy  is  expended  by  the  current,  i,  in  the  resistance,  r. 

3.)  The  power  equation  :  P0  =  ei,  where  P0  is  the 
power  expended  in  the  circuit  of  E.M.F.,  e,  and  current,  /". 

4.)    Kirchhoff's  laws  : 

a.)  The  sum  of  all  the  E.M.Fs.  in  a  closed  circuit  =  0, 
if  the  E.M.F.  consumed  by  the  resistance,  zr,  is  also  con- 
sidered as  a  counter  E.M.F.,  and  all  the  E.M.Fs.  are  taken 
in  their  proper  direction. 

b.)  The  sum  of  all  the  currents  flowing  towards  a  dis- 
tributing point  =  0. 

In  alternating-current  circuits,  that  is,  in  circuits  con- 
veying currents  which  rapidly  and  periodically  change  their 


2        fc '  v,Vt  AL  TJLRATA  KING-CURRENT  PHENOMENA.  [  §  2 

direction,  these  laws  cease  to  hold.  Energy  is  expended, 
not  only  in  the  conductor  through  its  ohmic  resistance,  but 
also  outside  of  it ;  energy  is  -stored  up  and  returned,  so 
that  large  currents  may  flow,  impressed  by  high  E.M.Fs., 
without  representing  any  considerable  amount  of  expended 
energy,  but  merely  a  surging  to  and  fro  of  energy ;  the 
ohmic  resistance  ceases  to  be  the  determining  factor  of 
current  strength ;  currents  may  divide  into  components, 
each  of  which  is  larger  than  the  undivided  current,  etc. 

2.  In  place  of  the  above-mentioned  fundamental  laws  of 
continuous  currents,  we  find  in  alternating-current  circuits 
the  following  : 

Ohm's  law  assumes  the  form,  i  =  e  j  z,  where  z,  the 
apparent  resistance,  or  impedance,  is  no  longer  a  constant 
of  the  circuit,  but  depends  upon  the  frequency  of  the  cur- 
rents ;  and  in  circuits  containing  iron,  etc.,  also  upon  the 
E.M.F. 

Impedance,  zt  is,  in  the  system  of  absolute  units,  of  the 
same  dimensions  as  resistance  (that  is,  of  the  dimension 
LT~l  =  velocity),  and  is  expressed  in  ohms. 

It  consists  of  two  components,  the  resistance,  r,  and  the 

reactance,  *r,  or  — 

z=  v r'2  -j-  x*. 

The  resistance,  r,  in  circuits  where  energy  is  expended 
only  in  heating  the  conductor,  is  the  same  as  the  ohmic 
resistance  of  continuous-current  circuits.  In  circuits,  how- 
ever, where  energy  is  also  expended  outside  of  the  con- 
ductor by  magnetic  hysteresis,  mutual  inductance,  dielectric 
hysteresis,  etc.,  r  is  larger  than  the  true  ohmic  resistance 
of  the  conductor,  since  it  refers  to  the  total  expenditure  of 
energy.  It  may  be  called  then  the  effective  resistance.  It 
is  no  longer  a  constant  of  the  circuit. 

The  reactance,  xt  does  not  represent  the  expenditure  of 
power,  as  does  the  effective  resistance,  r,  but  merely  the 
surging  to  and  fro  of  energy.  It  is  not  a  constant  of  the 


§3]  INTRODUCTION.  3 

circuit,  but  depends  upon  the  frequency,  and  frequently, 
as  in  circuits  containing  iron,  or  in  electrolytic  conductors, 
upon  the  E.M.F.  also.  Hence,  while  the  effective  resist- 
ance, r,  refers  to  the  ei^gy  component  of  E.M.F.,  or  the 
E.M.F.  in  phase  with  the^current,  the  reactance,  x,  refers 
to  the  wattless  component  of  E.M.F.,  or  the  E.M.F.  in 
quadrature  with  the  current. 

3.  The  principal  sources  of  reactance  are  electro-mag- 
netism and  capacity. 

ELECTRO-MAGNETISM. 

An  electric  current,  i,  flowing  through  a  circuit,  produces 
a  magnetic  flux  surrounding  the  conductor  in  lines  of 
magnetic  force  (or  more  correctly,  lines  of  magnetic  induc- 
tion), of  closed,  circular,  or  other  form,  which  alternate 
with  the  alternations  of  the  current,  and  thereby  induce 
an  E.M.F.  in  the  conductor.  Since  the  magnetic  flux  is 
in  phase  with  the  current,  and  the  induced  E.M.F.  90°,  or 
a  quarter  period,  behind  the  flux,  this  E.M.F.  of  self-induc- 
tance lags  90°,  or  a  quarter  period,  behind  the  current ;  that 
is,  is  in  quadrature  therewith,  and  therefore  wattless. 

If  now  4>  =  the  magnetic  flux  produced  by,  and  inter- 
linked with,  the  current  i  (where  those  lines  of  magnetic 
force,  which  are  interlinked  w-fold,  or  pass  around  n  turns 
of  the  conductor,  are  counted  n  times),  the  ratio,  <j>  /  iy  is 
denoted  by  Z,  and  called  self-inductance,  or  the  coefficient  of 
self-induction  of  the  circuit.  It  is  numerically  equal,  in 
absolute  units,  to  the  interlinkages  of  the  circuit  with  the 
magnetic  flux  produced  by  unit  current,  and  is,  in  the 
system  of  absolute  units,  of  the  dimension  of  length.  In- 
stead of  the  self-inductance,  Z,  sometimes  its  ratio  with 
the  ohmic  resistance,  r,  is  used,  and  is  called  the  Time- 
Constant  of  the  circuit  : 


4  ALTERNATING-CURRENT  PHENOMENA.  [§3 

If  a  conductor  surrounds  with  n  turns  a  magnetic  cir- 
cuit of  reluctance,  (R,  the  current,  z,  in  the  conductor  repre- 
sents the  M.M.F.  of  ni  ampere-turns,  and  hence  produces 
a  magnetic  flux  of  #z'/(R  lines  of  magnetic  force,  sur- 
rounding each  n  turns  of  the  conductor,  and  thereby  giving 
3>  =  ;/2z/(R  interlinkages  between  the  magnetic  and  electric 
circuits.  Hence  the  inductance  is  L  =  $/  i  =  n2  /  (R. 

The  fundamental  law  of  electro-magnetic  induction  is, 
that  the  E.M.F.  induced  in  a  conductor  by  a  varying  mag- 
netic field  is  the  rate  of  cutting  of  the  conductor  through 
the  magnetic  field. 

Hence,  if  i  is  the  current,  and  L  is  the  inductance  of 
a  circuit,  the  magnetic  flux  interlinked  with  a  circuit  of 
current,  i,  is  Li,  and  4  NLi  is  consequently  the  average 
rate  of  cutting  ;  that  is,  the  number  of  lines  of  force  cut 
by  the  conductor  per  second,  where  N  '  =  frequency,  or 
number  of  complete  periods  (double  reversals)  of  the  cur- 
rent per  second. 

Since  the  maximum  rate  of  cutting  bears  to  the  average 
rate  the  same  ratio  as  the  quadrant  to  the  radius  of  a 
circle  (a  sinusoidal  variation  supposed),  that  is  the  ratio 
TT  /  2  -7-  1,  the  maximum  rate  of  cutting  is  2  TT  TV,  and,  conse- 
quently, the  maximum  value  of  E.M.F.  induced  in  a  cir- 
cuit of  maximum  current  strength,  i,  and  inductance,  L,  is, 


Since  the  maximum  values  of  sine  waves  are  proportional 
(by  factor  V2)  to  the  effective  values  (square  root  of  mean 
squares),  if  /  =  effective  value  of  alternating  current,  e  = 
2  TT  NLi  is  the  effective  value  of  E.M.F.  of  self-inductance, 
and  the  ratio,  e  j  i  =  2  TT  NL,  is  the  magnetic  reactance  : 

xm  =  2  TT  NL.  m 
Thus,  if  r  =  resistance,  xm  =  reactance,  z  —  impedance,— 

the  E.M.F.  consumed  by  resistance  is  :  ^  =  ir  ; 
the  E.M.F.  consumed  by  reactance  is  :  e2  =  ioc^ 


§4]  INTRODUCTION.  5 

•and,   since  both  E.M.Fs.  are  in  quadrature  to  each  other, 
the  total  E.M.F.  is  — 


e  = 

that  is,  the  impedance,  ^takes  in  alternating-current  cir- 
cuits the  place  of  the  resistance,  r,  in  continuous-current 
circuits. 

CAPACITY. 

4.  If  upon  a  condenser  of  capacity,  C,  an  E.M.F.,  e,  is 
impressed,  the  condenser  receives  the  electrostatic  charge,  Ce. 

If  the  E.M.F.,  e,  alternates  with  the  frequency,  Nt  the 
•average  rate  of  charge  and  discharge  is  4  N,  and  2  rr  N  the 
maximum  rate  of  charge  and  discharge,  sinusoidal  waves  sup- 
posed, hence,  i  =  2  ?r  NCe  the  current  passing  into  the  con- 
denser, which  is  in  quadrature  to  the  E.M.F.,  and  leading. 


It  is  then:-        ^  =  7  =  2 

the  capacity  reactance,  or  condensance. 

Polarization  in  electrolytic  conductors  acts  to  a  certain 
extent  like  capacity. 

The  capacity  reactance  is  inversely  proportional  to  the 
frequency,  and  represents  the  leading  out-of-phase  wave  ; 
the  magnetic  reactance  is  directly  proportional  to  the 
frequency,  and  represents  the  lagging  out-of-phase  wave. 
Hence  both  are  of  opposite  sign  with  regard  to  each  other, 
and  the  total  reactance  of  the  circuit  is  their  difference, 
x  =  xm  xc  . 

The  total  resistance  of  a  circuit  is  equal  to  the  sum  of 
all  the  resistances  connected  in  series  ;  the  total  reactance 
of  a  circuit  is  equal  to  the  algebraic  sum  of  all  the  reac- 
tances connected  in  series  ;  the  total  impedance  of  a  circuit, 
however,  is  not  equal  to  the  sum  of  all  the  individual 
impedances,  but  in  general  less,  and  is  the  resultant  of  the 
total  resistance  and  the  total  reactance.  Hence  it  is  not 
permissible  directly  to  add  impedances,  as  it  is  with  resist- 
ances or  reactances. 


6  AL  TERN  A  TING-  CURRENT  PHENOMENA.       [§§5,6- 

A  further  discussion  of  these  quantities  will  be  found  in 
the  later  chapters. 

5.  In   Joule's    law,   P  =  fir,    r  is   not   the  true  ohmic 
resistance  any  more,   but  the  "effective  resistance;"  that 
is,  the  ratio  of  the  energy  component  of  E.M.F.  to  the  cur- 
rent.     Since  in  alternating-current  circuits,  besides  by  the 
ohmic    resistance    of    the    conductor,   energy   is    expended, 
partly    outside,    partly   even   inside,    of    the   conductor,   by 
magnetic  hysteresis,  mutual  inductance,  dielectric  hystere- 
sis, etc.,  the  effective  resistance,  r,  is  in  general  larger  than 
the  true  resistance  of  the  conductor,  sometimes  many  times 
larger,  as  in  transformers  at  open  secondary  circuit,  and  is 
not  a  constant  of  the  circuit  any  more.      It  is  more  fully 
discussed  in  Chapter  VII. 

In  alternating-current  circuits,  the  power  equation  con- 
tains a  third  term,  which,  in  sine  waves,  is  the  cosine  of 
the  difference  of  phase  between  E.M.F.  and  current  :  — 

P0  =  ei  cos  <£. 

Consequently,  even  if  e  and  i  are  both  large,  P0  may  be 
very  small,  if  cos  <£  is  small,  that  is,  </>  near  90°. 

Kirchhoff's  laws  become  meaningless  in  their  original 
form,  since  these  laws  consider  the  E.M.Fs.  and  currents 
as  directional  quantities,  counted  positive  in  the  one,  nega- 
tive in  the  opposite  direction,  while  the  alternating  current 
has  no  definite  direction  of  its  own. 

6.  The    alternating  waves    may   have  widely   different 
shapes ;    some    of  the   more   frequent    ones   are   shown   in 
a  later  chapter. 

The  simplest  form,  however,  is  the  sine  wave,  shown  in 
Fig.  1,  or,  at  least,  a  wave  very  near  sine  shape,  which 
may  be  represented  analytically  by  :  - 

i  =  /  sin  —  (/  -  A)  =  /sin  2  TT N  (t  -  A)  ; 


§6] 


INTRODUCTION. 


where  /  is  the  maximum  value  of  the  wave,  or  its  ampli- 
tude ;  T  is  the  time  of  one  complete  cyclic  repetition,  or 
the  period  of  the  wave,  or  N  =  1  /  T  is  the  frequency  or 
number  of  complete  pertods  per  second  ;  and  ^  is  the  time, 
where  the  wave  isy  zero,  o^the  epocJi  of  the  wave,  generally 
called  the  phase* 

Obviously,  "phase"  or  "epoch"  attains  a  practical 
meaning  only  when  several  waves  of  different  phases  are 
considered,  as  "difference  of  phase."  When  dealing  with 
one  wave  only,  we  may  count  the  time  from  the  moment 


f 

AN, 

\ 

'/ 

"~N 

\ 

/ 

c 

\ 

/ 

\ 

/ 

\ 

\ 

1 

\ 

/ 

5 

A 

V 

i 

N 

¥\ 

/ 

\ 

/ 

\ 

J 

\ 

v. 

2 

\ 

^ 

Fig.  1.    Sine  Waue. 

where  the  wave  is  zero,  or  from  the  moment   of  its  maxi- 
mum, and  then  represent  it  by  :  — 

i  =  I  sin  2  TT  Nt  • 
or,  /  =  /cos  2  TT  Nt. 

Since  it  is  a  univalent  function  of  time,  that  is,  can  at  a 
given  instant  have  one  value  only,  by  Fourier's  theorem, 
any  alternating  wave,  no  matter  what  its  shape  may  be, 
can  be  represented  by  a  series  of  sine  functions  of  different 
frequencies  and  different  phases,  in  the  form  :  — 

/  =  A  sin  2  TtN(t  —  /i)  +  L  sin  ±irN(t—  /2) 
7  sin  6 


*  "  Epoch  "  is  the  time  where  a  periodic  function  reaches  a  certain  value,. 
for  instance,  zero;  and  "phase"  is  the  angular  position,  with  respect  to  a 
datum  position,  of  a  periodic  function  at  a  given  time.  Both  are  in  alternate- 
current  phenomena  only  different  ways  of  expressing  the  same  thing. 


8 


ALTERNATING-CURRENT  PHENOMENA. 


[§6 


where  7j,  72,  73,  .  .  .  are  the  maximum  values  of  the  differ- 
ent components  of  the  wave,  t^  /2,  /3  .  .  .  the  times,  where 
the  respective  components  pass  the  zero  value. 

The  first  term,  7X  sin  2  TT  N  (t  —  tj,  is  called  the  fun- 
damental wave,  or  \.}\Q  first  harmonic;  the  further  terms  are 
called  the  higher  harmonics,  or  "overtones,"  in  analogy  to 
the  overtones  of  sound  waves.  7W  sin  2  mr  N  (/  —  /„)  is  the 
«th  harmonic. 

By  resolving  the  sine  functions  of  the  time  differences, 
/  —  /j,  t  —  /2  .  .  .  ,  we  reduce  the  general  expression  of 
the  wave  to  the  form  : 

i  =  ^i  sin  2  irNt  -f  ^2  sin  4  TT^V?  +  Az  sin  6  TrTV7?  +  .  .  . 
+  £l  cos  2  -xNt  +  A  cos  4  vNt  +  ^8  cos  6  irNt  +.  .  .  . 


/7g.  2.     Wave  without  Even  Harmonics. 

The  two  half-waves  of  each  period,  the  positive  wave 
and  the  negative  wave  (counting  in  a  definite  direction  in 
the  circuit),  are  almost  always  identical.  Hence  the  even 
higher  harmonics,  which  cause  a  difference  in  the  shape  of 
the  two  half-waves,  disappear,  and  only  the  odd  harmonics 
exist,  except  in  very  special  cases. 

Hence  the  general  alternating  current  is  expressed  by  : 

/=71sin27r^(/  —  A)  +  73  sin  §  TT  N  (t  —  /3) 


5  sin  10  TT  Nt  + 
$  cos  10  irNt  + 


or, 

i  =  A[  sin  2  TT  Nt  +  As  sin  6  irNt  + 
-f  B±  cos  2  irNt  -f  Bz  cos  6  -xNt  + 


§7] 


INTR  OD  UC  TION. 


9 


Such  a  wave  is  shown  in  Fig.  2,  while  Fig.  3  shows  a 
wave  whose  half-waves  are  different.  Figs.  2  and  3  repre- 
sent the  secondary  currents  of  a  Ruhmkorff  coil,  whose 
secondary  coil  is  closed  by"a  high  external  resistance  :  Fig. 
3  is  the  coil  operated  in  thfe  usual  way,  by  make  and  break 
of  the  primary  battery  current ;  Fig.  2  is  the  coil  fed  with 
reversed  currents  by  a  commutator  from  a  battery. 

7.  Self-inductance,  or  electro-magnetic  momentum,  which 
is  always  present  in  alternating-current  circuits,  —  to  a 
large  extent  in  generators,  transformers,  etc.,  —  tends  to. 


Fig..  3.     Wave  with  Even  Harmonics. 

suppress  the  higher  harmonics  of  a  complex  harmonic  wave 
more  than  the  fundamental  harmonic,  and  thereby  causes, 
a  general  tendency  towards  simple  sine  shape,  which  has 
the  effect,  that,  in  general,  the  alternating  currents  in  our 
light  and  power  circuits  are  sufficiently  near  sine  waves. 
to  make  the  assumption  of  sine  shape  permissible. 

Hence,  in  the  calculation  of  alternating-current  phe- 
nomena, we  can  safely  assume  the  alternating  wave  as  a 
sine  wave,  without  making  any  serious  error  ;  and  it  will  be 
sufficient  to  keep  the  distortion  from  sine  shape  in  mind 
as  a  possible,  though  improbable,  disturbing  factor,  which 


10  ALTERNATING-CURRENT  PHENOMENA.  [§7 

generally,  however,  is  in  practice  negligible  —  perhaps  with 
the  only  exception  of  low-resistance  circuits  containing  large 
magnetic  reactance,  and  large  condensances  in  series  with 
each  other,  so  as  to  produce  resonance  effects  of  these 
higher  harmonics. 


-•§8] 


INSTANTANEOUS  AND   INTEGRAL    VALUES. 


11 


»     CH 


APTK 


R    II 


INSTANTANEOUS  VALUES  AND  INTEGRAL  VALUES. 

8.  IN  a  periodically  varying  function,  as  an  alternating 
current,  we  have  to  distinguish  between  the  instantaneous 
value,  which  varies  constantly  as  function  of  the  time,  and 
the  integral  vahte,  which  characterizes  the  wave  as  a  whole. 

As  such  integral  value,  almost  exclusively  the  effective 


i 


Fig.  4.    Alternating  Wave. 

•vahte  is  used,  that  is,  the  square  root  of  the  mean  squares  ; 
and  wherever  the  intensity  of  an  electric  wave  is  mentioned 
without  further  reference,  the  effective  value  is  understood. 

The  maximum  value  of  the  wave  is  of  practical  interest 
only  in  few  cases,  and  may,  besides,  be  different  for  the  two 
half-waves,  as  in  Fig.  3. 

As  arithmetic  mean,  or  average  value,  of  a  wave  as  in 
Figs.  4  and  5,  the  arithmetical  average  of  all  the  instan- 
taneous values  during  one  complete  period  is  understood. 

This  arithmetic  mean  is  either  =  0,  as  in  Fig.  4,  or  it 
differs  from  0,  as  in  Fig.  5.  In  the  first  cas"e,  the  wave 
is  called  an  alternating  wave,  in  the  latter  a  pulsating  ^vave. 


12 


ALTERNATING-CURRENT  PHENOMENA. 


[§S 


Thus,  an  alternating  wave  is  a  wave  whose  positive 
values  give  the  same  sum  total  as  the  negative  values  ;  that 
is,  whose  two  half-waves  have  in  rectangular  coordinates 
the  same  area,  as  shown  in  Fig.  4. 

A  pulsating  wave  is  a  wave  in  which  one  of  the  half- 
waves  preponderates,  as  in  Fig.  5. 

Pulsating  waves  are  produced  only  by  commutating 
machines,  and  by  unipolar  machines  (or  by  the  superposi- 
tion of  alternating  waves  upon  continuous  currents,  etc.). 

All  inductive  apparatus  without  commutation  give  ex- 
clusively alternating  waves,  because,  no  matter  what  con- 


/ 

s**~ 

X, 

\ 

/ 

/ 

x 

N 

/ 

' 

X 

\ 

/ 

f 

/ 

\ 

7 

/ 

\ 

A 

VEF 

AQE 

VA 

LUE 

[ 

/ 

0 

\ 

1 

/ 

I 

/ 

\ 

s. 

/ 

\ 

"v* 

+S 

/ 

Fig.  5.    Pulsating  Wave. 

ditions  may  exist  in  the  circuit,  any  line  of  magnetic  force> 
which  during  a  complete  period  is  cut  by  the  circuit,  and 
thereby  induces  an  E.M.F.,  must  during  the  same  period 
be  cut  again  in  the  opposite  direction,  and  thereby  induce 
the  same  total  amount  of  E.M.F.  (Obviously,  this  does 
not  apply  to  circuits  consisting  of  different  parts  movable 
with  regard  to  each  other,  as  in  unipolar  machines.) 

In  the  following  we  shall  almost  exclusively  consider  the 
alternating  wave,  that  is  the  wave  whose  true  arithmetic 
mean  value  =  0. 

Frequently,  by  mean  value  of  an  alternating  wave,  the 
average  of  one  half-wave  only  is  denoted,  or  rather  the 


§9] 


INSTANTANEOUS  AND  INTEGRAL    VALUES. 


13 


average  of  all  instantaneous  values  without  regard  to  their 
sign.  This  mean  value  is  of  no  practical  importance,  and 
is,  besides,  in  many  cases  indefinite. 

9.    In  a  sine  wave,  the  relation  of  the  mean  to  the  maxi- 
mum value  is  found  in  the  following  way :  — 


Fig.  8. 

Let,  in  Fig.  6,  AOB  represent  a  quadrant  of  a  circle 
with  radius  1. 

Then,  while  the  angle  <£  traverses  the  arc  ?r  /  2  from  A  to 
B,  the  sine  varies  from  0  to  OB  =  1.  Hence  the  average 
variation  of  the  sine  bears  to  that  of  the  corresponding  arc 
the  ratio  1  -5-  ir/2,  or  2/7r  -5-  1.  The  maximum  variation 
of  the  sine  takes  place  about  its  zero  value,  where  the  sine 
is  equal  to  the  arc.  Hence  the  maximum  variation  of  the 
sine  is  equal  to  the  variation  of  the  corresponding  arc,  and 
consequently  the  maximum  variation  of  the  sine  bears  to 
its  average  variation  the  same  ratio  as  the  average  variation 
of  the  arc  to  that  of  the  sine ;  that  is,  1  -=-  2  /  ^,  and  since 
the  variations  of  a  sine-function  are  sinusoidal  also,  we 
have, 


Mean  value  of  sine  wave 


maximum  value  = hi 

7T 

=  .63663. 


The  quantities,  "current,"  "E.M.F.,"  "magnetism,"  etc., 
are  in  reality  mathematical  fictions  only,  as  the  components 


14 


ALTERNATING-CURRENT  PHENOMENA. 


[§9 


of  the  entities,  "energy,"  "power,"  etc. ;  that  is,  they  have 
no  independent  existence,  but  appear  only  as  squares  or 
products. 

Consequently,  the  only  integral  value  of  an  alternating 
wave  which  is  of  practical  importance,  as  directly  connected 
with  the  mechanical  system  of  units,  is  that  value  which 
represents  the  same  power  or  effect  as  the  periodical  wave. 
This  is  called  the  effective  value.  Its  square  is  equal  to  the 
mean  square  of  the  periodic  function,  that  is  :  — 

The  effective  value  of  an  alternating  wave,  or  the  value 
representing  the  same  effect  as  the  periodically  varying  wave, 
is  the  square  root  of  the  mean  square. 

In  a  sine  wave,  its  relation  to  the  maximum  value  is 
found  in  the  following  way : 


Fig.  7. 


Let,  in  Fig.  7,  AOB  represent  a  quadrant  of  a  circle 
with  radius  1. 

Then,  since  the  sines  of  any  angle  <j>  and  its  complemen- 
tary angle,  90°—  <£,  fulfill  the  condition, — 

sin2  <£  +  sin2  (90  —  <£)=!, 

the  sines  in  the  quadrant,  AOB,  can  be  grouped  into  pairs, 
so  that  the  sum  of  the  squares  of  any  pair  =  1 ;  or,  in  other 
words,  the  mean  square  of  the  sine  =1/2,  and  the  square 
root  of  the  mean  square,  or  the  effective  value  of  the  sine, 
=  1  /  V2.  That  is  : 


:§  1O]       INSTANTANEOUS  AND   INTEGRAL    VALUES. 


15 


The  effective  value  of  a  sine  function  bears  to  its  maxi- 
mum value  the  ratio,  — 

A=  -r  1  =  -70711. 

Hence,  we  have  for  trfe  sine  curve  the  following  rela- 
tions : 


MAX. 

EFF. 

ARITH.  MEAN. 

Half 
Period. 

Whole 
Period. 

1 

1 
V2 

2 
7T 

0 

1 

.7071 

.63663 

0 

1.4142 

1 

.90034 

0 

1.5708 

1.1107 

1 

0 

10.    Coming  now  to  the  general  alternating  curve, — 

i  =  ^  sin  27r  Nt  +  Az  sin  4rr  Nt  +  A*  sin  6ir  Nt  +  .  .  . 
+  .#!  cos  lieNt  +  BZ  cos  47r7V?  +  ^3  cos  GTT^W  +  .  .  . , 

we  find,  by  squaring  this  expression  and  canceling  all  the 
products  which  give  0  as  mean  square,  the  effective  vahie,  — 

The  mean  value  does  not  give  a  simple  expression,  and 
is  of  no  general  interest. 


16  AL  TERN  A  TING-CURRENT  PHENOMENA.          [§11 


CHAPTER    III. 

LAW   OF    ELECTRO-MAGNETIC    INDUCTION. 

11.  If  an  electric  conductor  moves  relatively  to  a  mag- 
netic field,  an  E.M.F.  is  induced  in  the  conductor  which  is 
proportional  to  the  intensity  of  the  magnetic  field,  to  the 
length  of  the  conductor,  and  to  the  speed  of  its  motion 
perpendicular  to  the  magnetic  field  and  the  direction  of  the 
conductor ;  or,  in  other  words,  proportional  to  the  number 
of  lines  of  magnetic  force  cut  per  second  by  the  conductor. 

As  a  practical  unit  of  E.M.F.,  the  volt  is  defined  as  the 
E.M.F.  induced  in  a  conductor,  which  cuts  108  =  100,000,000 
lines  of  magnetic  force  per  second. 

If  the  conductor  is  closed  upon  itself,  the  induced  E.M.F. 
produces  a  current. 

A  closed  conductor  may  be  called  a  turn  or  a  convolution. 
In  such  a  turn,  the  number  of  lines  of  magnetic  force  cut 
per  second  is  the  increase  or  decrease  of  the  number  of 
lines  inclosed  by  the  turn,  or  ;/  times  as  large  with  ;/  turns. 

Hence  the  E.M.F.  in  volts  induced  in  n  turns,  or  con- 
volutions, is  n  times  the  increase  or  decrease,  per  second, 
of  the  flux  inclosed  by  the  turns,  times  10~8. 

If  the  change  of  the  flux  inclosed  by  the  turn,  or  by  n 
turns,  does  not  take  place  uniformly,  the  product  of  the 
number  of  turns,  times  change  of  flux  per  second,  gives 
the  average  E.M.F. 

If  the  magnetic  flux,  4>,  alternates  relatively  to  a  number 
of  turns,  n  —  that  is,  when  the  turns  either  revolve  through 
the  flux,  or  the  flux  passes  in  and  out  of  the  turns,  during 
each  complete  alternation  or  cycle,  —  the  total  flux  is  cut 
four  times,  twice  passing  into,  and  twice  out  of,  the  turns. 


§12]       LA  W  OF  ELECTRO-MA  GNE  TIC  2ND UCTION.  1 7 

Hence,  if  N  =  number  of  complete  cycles  per  second, 
or  the  frequency  of  the  relative  alternation  of  flux  <£,  the 
average  E.M.F.  induced  In  n  turns  is, — 

^avg.  =  ±?i®N  10 -«  volts. 

This  is  the  fundamental  equation  of  electrical  engineer- 
ing, and  applies  to  continuous-current,  as  well  as  to  alter- 
nating-current, apparatus. 

12.  In  continuous-current  machines  and  in  many  alter- 
nators, the  turns  revolve  through  a  constant  magnetic 
field ;  in  other  alternators  and  in  induction  motors,  the  mag- 
netic field  revolves ;  in  transformers,  the  field  alternates 
with  respect  to  the  stationary  turns. 

Thus,  in  the  continuous-current  machine,  if  n  =  num- 
ber of  turns  in  series  from  brush  to  brush,  <l>  =  flux  inclosed 
per  turn,  and  N  =  frequency,  the  E.M.F.  induced  in  the 
machine  is  E  =  4;z3>^V10~8  volts,  independent  of  the  num- 
ber of  poles,  or  series  or  multiple  connection  of  the  arma- 
ture, whether  of  the  ring,  drum,  or  other  type. 

In  an  alternator  or  transformer,  if  n  is  the  number  of 
turns  in  series,  <$  the  maximum  flux  inclosed  per  turn,  and 
N  the  frequency,  this  formula  gives, 

£&VK  =  4  n$>  JV10  -8  volts. 
Since  the  maximum  E.M.F.  is  given  by,  — 

^max.  =  |  avg.  E, 
we  have 

^ma*.   =  2  7T  ff  <D  N  10  ~  8  VOltS. 

And  since  the  effective  E.M.F.  is  given  by, — 


we  have 


4.44  n  3>  .AT  10  ~8  volts, 


which   is    the    fundamental    formula  of    alternating-current 
induction  by  sine  waves. 


18  AL  TERN  A  TING-CURRENT  PHENOMENA .          [§13 

13.  If,  in  a  circuit  of  n  turns,  the  magnetic  flux,  O, 
inclosed  by  the  circuit  is  produced  by  the  current  flowing 
in  the  circuit,  the  ratio  — 

flux  X  number  of  turns  X  10~8 
current 

is  called  the  inductance,  L,  of  the  circuit,  in  henrys. 

The  product  of  the  number  of  turns,  n,  into  the  maxi- 
mum flux,  <l>,  produced  by  a  current  of  /amperes  effective,. 
or/V2  amperes  maximum,  is  therefore  — 

n$>  =Z/V2  108; 
and  consequently  the  effective  E.M.F.  of  self -inductance  is: 

E  =  V27r/z4>^10-8 
=  2-,rNLI  volts. 

The  product,  x  =  2  WVZ,  is  of  the  dimension  of  resistance, 
and  is  called  the  reactance  of  the  circuit ;  and  the  E.M.F. 
of  self-inductance  of  the  circuit,  or  the  reactance  voltage,  is 

E  =  Ix, 

and  lags  90°  behind  the  current,  since  the  current  is  in 
phase  with  the  magnetic  flux  produced  by  the  current, 
and  the  E.M.F.  lags  90°  behind  the  magnetic  flux.  The 
E.M.F.  lags  90°  behind  the  magnetic  flux,  as  it  is  propor- 
tional to  the  change  in  flux  ;  thus  it  is  zero  when  the  mag- 
netism is  at  its  maximum  value,  and  a  maximum  when  the 
flux  passes  through  zero,  where  it  changes  quickest. 


§14] 


GRAPHIC  REPRESENTA  TION. 


CHAPTER    IV. 


GRAPHIC    REPRESENTATION. 

14.  While  alternating  waves  can  be,  and  frequently  are, 
represented  graphically  in  rectangular  coordinates,  with  the 
time  as  abscissae,  and  the  instantaneous  values  of  the  wave 
as  ordinates,  the  best  insight  with  regard  to  the  mutual 
relation  of  different  alternate  waves  is  given  by  their  repre- 
sentation in  polar  coordinates,  with  the  time  as  an  angle  or 
the  amplitude,  —  one  complete  period  being  represented  by 
one  revolution,  —  and  the  instantaneous  values  as  radii 
vectores. 


Fig.  8. 


Thus  the  two  waves  of  Figs.  2  and  3  are  represented  in 
polar  coordinates  in  Figs.  8  and  9  as  closed  characteristic 
curves,  which,  by  their  intersection  with  the  radius  vector, 
give  the  instantaneous  value  of  the  wave,  corresponding  to 
the  time  represented  by  the  amplitude,  of  the  radius  vector. 

These  instantaneous  values  are  positive  if  in  the  direction 
of  the  radius  vector,  and  negative  if  in  opposition.  Hence 
the  two  half-waves  in  Fig.  1-  are  represented  by  the  same 


20 


AL  TERNA  TING-CURRENT  PHENOMENA. 


[§15 


polar  characteristic  curve,  which  is  traversed  by  the  point  of 
intersection  of  the  radius  vector  twice  per  period,  —  once 
in  the  direction  of  the  vector,  giving  the  positive  half-wave, 


Fig.  9. 


and  once  in  opposition  to  the  vector,  giving  the  negative 
half-wave.  In  Figs.  3  and  9,  where  the  two  half-waves  are 
different,  they  give  different  polar  characteristics. 


Fig.  10. 


15.  The  sine  wave,  Fig.  1,  is  represented  in  polar 
coordinates  by  one.  circle,  as  shown  in  Fig.  10.  The 
diameter  of  the  characteristic  curve  of  the  sine  wave, 
/=  OC,  represents  the  intensity  of  the  wave;  and  the  am- 
plitude of  the  diameter,  OC,  Z  w  =  AOC,  is  the  phase  of  the 


§16]  GRAPHIC  REPRESENTATION.  21 

wave,  which,  therefore,  is  represented  analytically  by  the 

function  :  — 

,,  i  —  I  cos  (  (j)  —  to), 

where  <f>  =  2TI7  /  T  is  the  instantaneous  value  of  the  ampli- 
tude corresponding  to  the  instantaneous  value,  i,  of  the  wave. 

The  instantaneous  values  are  cut  out  on  the  movable  ra- 
dius vector  by  its  intersection  with  the  characteristic  circle. 
Thus,  for  instance,  at  the  amplitude  AOBl  =  ^  =  2 irt-^ /  T 
(Fig.  10),  the  instantaneous  value  is  OB ';  at  the  amplitude 
AOB2  =  (/>2  =  27T/2  /  Tt  the  instantaneous  value  is  OB" ,  and 
negative,  since  in  opposition  to  the  vector  OB2. 

The  characteristic  circle  of  the  alternating  sine  wave  is 
determined  by  the  length  of  its  diameter  —  the  intensity 
of  the  wave,  and  by  the  amplitude  of  the  diameter  —  the 
phase  of  the  wave. 

Hence,  wherever  the  integral  value  of  the  wave  is  con- 
sidered alone,  and  not  the  instantaneous  values,  the  charac- 
teristic circle  may  be  omitted  altogether,  and  the  wave 
represented  in  intensity  and  in  phase  by  the  diameter  of 
the  characteristic  circle. 

Thus,  in  polar  coordinates,  the  alternate  wave  is  repre- 
sented in  intensity  and  phase  by  the  length  and  direction 
of  a  vector,  OC,  Fig.  10. 

Instead  of  the  maximum  value  of  the  wave,  the  effective 
value,  or  square  root  of  mean  square,  may  be  used  as  the 
vector,  which  is  more  convenient ;  and  the  maximum  value 
is  then  V2  times  the  vector  OC,  so  that  the  instantaneous 
values,  when  taken  from  the  diagram,  have  to  be  increased 
by  the  factor  V2. 

16.  To  combine  different  sine  waves,  their  graphical 
representations,  or  vectors,  are  combined  by  the  parallelo- 
gram law. 

If,  for  instance,  two  sine  waves,  OB  and  OC  (Fig.  11), 
are  superposed,  —  as,  for  instance,  two  E.M.Fs.  acting  in 
the  same  circuit,  —  their  resultant  wave  is  represented  by 


22 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


[§16 


ODy  the  diagonal  of  a  parallelogram  with  OB  and  OC  as. 
sides. 

For  at  any  time,  /,  represented  by  angle  <£  =  AOX,  trie- 
instantaneous  values  of  the  three  waves,  OB,  OC,  OD,  are 
their  projections  upon  OX,  and  the  sum  of  the  projections 
of  OB  and  OC  is  equal  to  the  projection  of  OD\  that  is,  the 
instantaneous  values  of  the  wave  OD  are  equal  to  the  sum 
of  the  instantaneous  values  of  waves  OB  and  OC. 

From  the  foregoing  considerations  we  have  the  con- 
clusions : 

The  sine  wave  is  represented  graphically  in  polar  coordi- 
nates by  a  vector,  which  by  its  lengthy  OCy  denotes  the  in- 


Fig.   11. 


tensity,  and  by  its  amplitude,  AOC,  the  phase,  of  the  sine 
wave. 

Sine  waves  are  combined  or  resolved  graphically,  in  polar 
coordinates,  by  the  law  of  parallelogram  or  the  polygon  of 
sine  waves. 

Kirchhoff's  laws  now  assume,  for  alternating  sine  waves,, 
the  form  :  — 

a.)  The  resultant  of  all  the  E.M.Fs.  in  a  closed  circuit,, 
as  found  by  the  parallelogram  of  sine  waves,  is  zero  if 
the  counter  E.M.Fs.  of  resistance  and  of  reactance  are 
included. 

b.)    The  resultant  of  all  the  currents  flowing  towards  a 


17] 


GRAPHIC  REPRESENTA  TION. 


23 


distributing  point,  as  found   by  the  parallelogram  of   sine 

waves,  is  zero. 

The  energy  equation  expressed  graphically  is  as  follows  : 
The  power   of    an   alternating-current   circuit    is   repre- 

sented in  polar  coordinates  by  the  product  of  the  current  ; 

/,  into  the  projection  of  the  E.M.F.,  E,  upon  the  current,  or 

by  the  E.M.F.,  E,  into  the  projection  of  the  current,  /,  upon 

the  E.M.F.,  or  IE  cos 


17.  Suppose,  as  an  instance,  that  over  a  line  having  the 
resistance,  r,  and  the  reactance,  x  —  QirNL,  —  where  N  = 
frequency  and  L  =  inductance,  —  a  current  of  /  amperes 
be  sent  into  a  non-inductive  circuit  at  an  E.M.F.  of  E 


Fig.   12. 


volts.     What  will  be  the  E.M.F.  required  at  the  generator 
end  of  the  line  ? 

In  the  polar  diagram,  Fig.  12,  let  the  phase  of  the  cur- 
rent be  assumed  as  the  initial  or  zero  line,  OL  Since  the 
receiving  circuit  is  non-inductive,  the  current  is  in  phase 
with  its  E.M.F.  Hence  the  E.M.F.,  E,  at  the  end  of  the 
line,  impressed  upon  the  receiving  circuit,  is  represented  by 
a  vector,  OE.  To  overcome  the  resistance,  r,  of  the  line, 
an  E.M.F.,  Ir,  is  required  in  phase  with  the  current,  repre- 
s'ented  by  OEr  in  the  diagram.  The  self-inductance  of  the 
line  induces  an  E.M.F.  which  is  proportional  to  the  current 
/  and  reactance  x,  and  lags  a  quarter  of  a  period,  or  90°,, 
behind  the  current.  To  overcome  this  counter  E.M.F, 


AL  TERNA  TING-CURRENT  PHENOMENA. 


18 


of  self-induction,  an  E.M.F.  of  the  value  Ix  is  required, 
in  phase  90°  ahead  of  the  current,  hence  represented  by 
vector  OEX.  Thus  resistance  consumes  E.M.F.  in  phase, 
and  reactance  an  E.M.F.  90°  ahead  of  the  current.  The 
E.M.F.  of  the  generator,  E0,  has  to  give  the  three  E.M.Fs., 
E,  Er,  and  Ex,  hence  it  is  determined  as  their  resultant. 
Combining  by  the  parallelogram  law,  OEr  and  OEX,  give 
OEZ,  the  E.M.F.  required  to  overcome  the  impedance  of 
the  line,  and  similarly  OEZ  and  OE  give  OE0,  the  E.M.F. 
required  at  the  generator  side  of  the  line,  to  yield  the 
E.M.F.  E  at  the  receiving  end  of  the  line.  Algebraically, 
we  get  from  Fig.  12  — 


or,  E   =    .      _ 

In  this  instance  we  have  considered  the  E.M.F.  con- 
sumed by  the  resistance  (in  phase  with  the  current)  and 
the  E.M.F.  consumed  by  the  reactance  (90°  ahead  of  the 
current)  as  parts,  or  components,  of  the  impressed  E.M.F., 
E0y  and  have  derived  E0  by  combining  Er,  Ex,  and  E. 


Fig.  13. 


18.  We  may,  however,  introduce  the  effect  of  the  induc- 
tance directly  as  an  E.M.F.,  Ex  ,  the  counter  E.M.F.  of 
self-induction  =  Ig,  and  lagging  90°  behind  the  current ;  and 
the  E.M.F.  consumed  by  the  resistance  as  a  counter  E.M.F., 
Ef  =  Ir,  but  in  opposition  to  the  current,  as  is  done  in  Fig. 
13  ;  and  combine  the  three  E.M.Fs.  E0,  EJ,  Ex,  to  form  a 
resultant  E.M.F.,  E,  which  is  left  at  the  end  of  the  line. 


§18] 


GRAPHIC  REPRESENTA  TION, 


E±  and  Ex'  combine  to  form  Ez',  the  counter  E.M.F.  of 
impedance ;  and  since  Eg  and  E0  must  combine  to  form 
E,  E0  is  found  as  the  side  of  a  parallelogram,  OE0EE^t 
whose  other  side,  OEZ,  and"  diagonal,  OE,  are  given. 

Or  we  may  say  (Fig.  14£,  that  to  overcome  the  counter 
E.M.F.  of  impedance,  OEZ',  of  the  line,  the  component,  OEer 
of  the  impressed  E.M.F.,  together  with  the  other  component 
OE,  must  give  the  impressed  E.M.F.,  OE0. 

As  shown,  we  can  represent  the  E.M.Fs.  produced  in  a 
circuit  in  two  ways  —  either  as  counter  E.M.Fs.,  which  com- 
bine with  the  impressed  E.M.F.,  or  as  parts,  or  components,. 


E.'r    O 


Fig.  14. 


of  the  impressed  E.M.F.,  in  the  latter  case  being  of  opposite 
phase.  According  to  the  nature  of  the  problem,  either  the 
one  or  the  other  way  may  be  preferable. 

As  an  example,  the  E.M.F.  consumed  by  the  resistance 
is  Ir,  and  in  phase  with  the  current  ;  the  counter  E.M.F. 
of  resistance  is  in  opposition  to. the  current.  The  E.M.F. 
consumed  by  the  reactance  is  Ix,  and  90°  ahead  of  the  cur- 
rent, while  the  counter  E.M.F.  of  reactance  is  90°  behind 
the  current ;  so  that,  if,  in  Fig.  15,  OI,  is  the  current,  — 

OEr    =  E.M.F.  consumed  by  resistance, 
OEr'  =  counter  E.M.F.  of  resistance, 
OEX    =  E.M.F.  consumed  by  inductance, 
OEX'  =  counter  E.M.F.  of  inductance, 
OEZ    =  E.M.F.  consumed  by  impedance, 
OEZ'  =  counter  E.M.F.  of  impedance. 


ALTERNATING-CURRENT  PHENOMENA.  [§§19,2O 


Obviously,  these  counter  E.M.Fs.  are  different  from,  for 
instance,  the  counter  E.M.F.  of  a  synchronous  motor,  in  so 
far  as  they  have  no  independent  existence,  but  exist  only 
through,  and  as  long  as,  the  current  flows.  In  this  respect 
they  are  analogous  to  the  opposing  force  of  friction  in 
mechanics. 


E'Z 


E'r     O 
Ex 


|Er 

Ei 


Fig.   15. 


19.    Coming  back  to  the  equation  found  for  the  E.M.F. 
at  the  generator  end  of  the  line,  — 


we  find,  as  the  drop  of  potential  in  the  line  — 


Irf 


E. 


This  is   different   from,   and   less   than,   the   E.M.F.  of 
impedance  — 


Ez  =  Iz  -- 

Hence  it  is  wrong  to  calculate  the  drop  of  potential  in  a 
circuit  by  multiplying  the  current  by  the  impedance ;  and  the 
drop  of  potential  in  the  line  depends,  with  a  given  current 
fed  over  the  line  into  a  non-inductive  circuit,  not  only  upon 
the  constants  of  the  line,  r  and  x,  but  also  upon  the  E.M.F., 
Ej  at  end  of  line,  as  can  readily  be  seen  from  the  diagrams. 

20.  If  the  receiver  circuit  is  inductive,  that  is,  if  the 
current,  /,  lags  behind  the  E.M.F.,  E,  by  an  angle  w,  and 
we  choose  again  as  the  zero  line,  the  current  OI  (Fig.  16), 
the  E.M.F.,  OE  is  ahead  of  the  current  by  angle  w.  The 


§20] 


GRAPHIC  REPRESENTA  TION, 


27 


E.M.F.  consumed  by  the  resistance,  Ir,  is  in  phase  with  the 
current,  and  represented  by  OEr\  the  E.M.F.  consumed 
by  the  reactance,  fx,  is  90°  ahead  of  the  current,  and  re- 
presented by  OEX.  Combining  OE,  OEri  and  OEX,  we 
get  OE0,  the  E.M.-F-.  required  at  the  generator  end  of  the 
line.  Comparing  Fig.  16  with  Fig.  13,  we  see  that  in 
the  former  OE0  is  larger;  or  conversely,  if  E0  is  the  same, 
E  will  be  less  with  an  inductive  load.  In  other  words, 
the  drop  of  potential  in  an  inductive  line  is  greater,  if  the 
receiving  circuit  is  inductive,  than  if  it  is  non-inductive. 
From  Fig.  16,  — 


E0  =  -V(J5  cos  w  +  fr)2  +  (E  sin  & 


Er 


Fig.  16. 

If,  however,  the  current  in  the  receiving  circuit  is 
leading,  as  is  the  case  when  feeding  condensers  or  syn- 
chronous motors  whose  counter  E.M.F.  is  larger  than  the 
impressed  E.M.F.,  then  the  E.M.F.  will  be  represented,  in 
Fig.  17,  by  a  vector,  OE,  lagging  behind  the  current,  Of, 
by  the  angle  of  lead  £';  and  in  this  case  we  get,  by 
•combining  OE  with  OEr,  in  phase  with  the  current,  and 
~OEX,  90°  ahead  of  the  current,  the  generator  E.M.F.,  OE~0, 
which  in  this  case  is  not  only  less  than  in  Fig.  16  and  in 
Fig.  13,  but  may  be  even  less  than  E ;  that  is,  the  poten- 
tial rises  in  the  line.  In  other  words,  in  a  circuit  with 
leading  current,  the  self-induction  of  the  line  raises  the 
potential,  so  that  the  drop  of  potential  is  less  than  with 


28 


AL  TERN  A  TING-  CURRENT  PHENOMENA. 


[§21 


a  non-inductive  load,  or  may  even  be  negative,  and  the 
voltage  at  the  generator  lower  than  at  the  other  end  of 
the  line. 

These  same  diagrams,  Figs.  13  to  17,  can  be  considered 
as  polar  diagrams  of  an  alternating-current  generator  of- 
an  E.M.F.,  E0,  a  resistance  E.M.F.,  Er  =  Ir,  a  reactance, 
Ex  —  Ix,  and  a  difference  of  potential,  E,  at  the  alternator 
terminals;  and  we  see,  in  this  case,  that  with  an  inductive 
load  the  potential  difference  at  the  alternator  terminals  will 
be  lower  than  with  a  non-inductive  load,  and  that  with  a 
non-inductive  load  it  will  be  lower  than  when  feeding  into 


Fig.  17. 


a  circuit  with  leading  current,  as,  for  instance,  a  synchro- 
nous motor  circuit  under  the  circumstances  stated  above. 

21.  As  a  further  example,  we  may  consider  the  dia- 
gram of  an  alternating-current  transformer,  feeding  through 
its  secondary  circuit  an  inductive  load. 

For  simplicity,  we  may  neglect  here  the  magnetic 
hysteresis,  the  effect  of  which  will  be  fully  treated  in  a 
separate  chapter  on  this  subject. 

Let  the  time  be  counted  from  the  moment  when  the 
magnetic  flux  is  zero.  The  phase  of  the  flux,  that  is,  the 
amplitude  of  its  maximum  value,  is  90°  in  this  case,  and, 
consequently,  the  phase  of  the  induced  E.M.F.,  is  180°,. 


§21 


GRAPHIC  REPRESENTA  TION. 


29 


since  the  induced  .  E.M.F.  lags  90°  behind  the  inducing 
flux.  Thus  the  secondary  induced  E.M.F.,  £lt  will  be 
represented  by  a  vector,  Q£lt  in  Fig.  18,  at  the  phase 
180°.  The  secondary  current,  Iv  lags  behind  the  E.M.F.  El 
by  an  angle  G>lf  which  is  determined  by  the  resistance  and 
inductance  of  the  secondary  circuit ;  that  is,  by  the  load  in 
the  secondary  circuit,  and  is  represented  in  the  diagram  by 
the  vector  OFlt  of  phase  180  -f  5. 


Fig.  18. 


Instead  of  the  secondary  current,  flt  we  plot,  however, 
the  secondary  M.M.F.,  Fl  =  n^  flt  where  «x  is  the  number 
of  secondary  turns,  and  F1  is  given  in  ampere-turns.  This 
makes  us  independent  of  the  ratio  of  transformation. 

From  the  secondary  induced  E.M.F.,  Elt  we  get  the  flux, 
3>,  required  to  induce  this  E.M.F.,  from  the  equation  — 


I 

where  — 

EI  =  secondary  induced  E.M.F.,  in  effective  volts, 

JV  —  frequency,  in  cycles  per  second. 

7/x    =  number  of  secondary  turns. 

<£    =  maximum  value  of  magnetic  flux,  in  webers. 

The   derivation  of  this  equation    has    been   given  in  a 
preceding  chapter. 

This  magnetic  flux,  <|>,  is  represented  by  a  vector,  O&,  at 


30  ALTERNATING-CURRENT  PHENOMENA.  [§22 

the  phase  90°,  and  to  induce  it  an  M.M.F.,  &,  is  required, 
which  is  determined  by  the  magnetic  characteristic  of  the 
iron,  and  the  section  and  length  of  the  magnetic  circuit  of 
the  transformer ;  it  is  in  phase  with  the  flux  <£,  and  repre- 
sented by  the  vector  OF,  in  effective  ampere-turns. 

The  effect  of  hysteresis,  neglected  at  present,  is  to  shift 
OF  ahead  of  OM,  by  an  angle  a,  the  angle  of  hysteretic 
lead.  (See  Chapter  on  Hysteresis.) 

This  M.M.F.,  &,  is  the  resultant  of  the  secondary  M.M.F., 
&lt  and  the  primary  M.M.F.,  &0\  or  graphically,  OF  is  the 
diagonal  of  a  parallelogram  with  OF^  and  OF0  as  sides.  OFl 
and  OF  being  known,  we  find  OF0,  the  primary  ampere- 
turns,  n0 ,  and  therefrom,  the  primary  current,  I0  =  $0  /  n0 , 
which  corresponds  to  the  secondary  current,  II. 

To  overcome  the  resistance,  r0,  of  the  primary  coil,  an 
E.M.F.,  Er  =  Ir0,  is  required,  in  phase  with  the  current,  70, 
and  represented  by  the  vector  OEr. 

To  overcome  the  reactance,  x0  =  2  TT  n0  L0 ,  of  the  pri- 
mary coil,  an  E.M.F.  Ex  =  I0x0  is  required,  90°  ahead  of 
the  current  70,  and  represented  by  vector,  OEX. 

The  resultant  magnetic  flux,  $,  which  in  the  secondary 
-coil  induces  the  E.M.F.,  £1,  induces  in  the  primary  coil  an 
E.M.F.  proportional  to  £1  by  the  ratio  of  turns  n0j  n^,  and 
in  phase  with  £l ,  or,  — 


which  is  represented  by  the  vector  OEt' .  To  overcome  this 
counter  E.M.F.,  E{',  a  primary  E.M.F.,  Elt  is  required,  equal 
but  opposite  to  £t' ' ,  and  represented  by  the  vector,  OEV. 

The  primary  impressed  E.M.F.,  E0,  must  thus  consist  of 
the  three  components,  OE^  OEr,  and  OEX,  and  is,  there- 
fore, a  resultant  OE0,  while  the  difference  of  phase  in  the 
primary  circuit  is  found  to  be  ^f  %  =  E0OA. 

22.  Thus,  in  Figs.  18,  19,  and  20,  the  diagram  of  an 
.alternating-current  transformer  is  drawn  for  the  same  sec- 


§22] 


GRAPHIC  REPRESENTATION. 


31 


ondary  E.M.F.,  Elt  and  secondary  current,  Ilt  but  with  dif- 
ferent conditions  of  secondary  displacement :  — 

In  Fig.  18,  the  secondary  current,  I± ,  lags  60°  behind  the  sec- 
ondary E.M.F.,  jffj.  . 

In  Fig.  19,  the  secondary  current,  fl9  is  in  phase  with  the 
secondary  E.M.F.,  JSl. 

In  Fig.  20,  the  secondary  current,  7i ,  leads  by  60°  the  second- 
ary E.M.F.,  EI. 


Fig.  19. 


These  diagrams  show  that  lag  in  the  secondary  circuit  in- 
creases and  lead  decreases,  the  primary  current  and  primary 
E.M.F.  required  to  produce  in  the  secondary  circuit  the 
,same  E.M.F.  and  current ;  or  conversely,  at  a  given  primary 


Fig.  20. 

impressed  E.M.F.,  E0,  the  secondary  E.M.F.,  Ev  will  be 
smaller  with  an  inductive,  and  larger  with  a  condenser 
(leading  current)  load,  than  with  a  non-inductive  load. 

At  the   same  time  we   see  that   a  difference  of  phase 
'existing  in  the  secondary  circuit  of  a  transformer  reappears 


32  ALTERNATING-CURRENT  PHENOMENA.  [§22 

in  the  primary  circuit,  somewhat  decreased  if  leading,  and 
slightly  increased  if  lagging.  Later  we  shall  see  that 
hysteresis  reduces  the  displacement  in  the  primary  circuit, 
so  that,  with  an  excessive  lag  in  the  secondary  circuit,  the 
lag  in  the  primary  circuit  may  be  less  than  in  the  secondary. 
A  conclusion  from  the  foregoing  is  that  the  transformer 
is  not  suitable  for  producing  currents  of  displaced  phase  \ 
since  primary  and  secondary  current  are,  except  at  very 
light  loads,  very  nearly  in  phase,  or  rather,  in  opposition, 
to  each  other. 


23]  SYMBOLIC  METHOD.  33 


CHAPTER    V. 

SYMBOLIC    METHOD. 

23.  The  graphical  method  of  representing  alternating- 
<current  phenomena  by  polar  coordinates  of  time  affords  the 
best  means  for  deriving  a  clear  insight  into  the  mutual  rela- 
tion of  the  different  alternating  sine  waves  entering  into  the 
problem.  For  numerical  calculation,  however,  the  graphical 
method  is  frequently  not  well  suited,  owing  to  the  widely 
different  magnitudes  of  the  alternating  sine  waves  repre- 
sented in  the  same  diagram,  which  make  an  exact  diagram- 
matic determination  impossible.  For  instance,  in  the  trans- 
former diagrams  (cf.  Figs.  18-20),  the  different  magnitudes 
will  have  numerical  values  in  practice,  somewhat  like  E^  = 
100  volts,  and  7X  =  75  amperes,  for  a  non-inductive  secon- 
dary load,  as  of  incandescent  lamps.  Thus  the  only  reac- 
tance of  the  secondary  circuit  is  that  of  the  secondary  coil, 
or,  x^  =  .08  ohms,  giving  a  lag  of  wx  =  3.6°.  We  have 

.also, 

n^   =      30  turns. 

n0   =    300  turns. 

$1  =  2250  ampere-turns. 

ff    =    100  ampere-turns. 

Er  =      10  volts. 

Ex  =      60  volts. 

Ei  =  1000  volts. 

The  corresponding  diagram  is  shown  in  Fig.  21.  Obvi- 
ously, no  exact  numerical  values  can  be  taken  from  a  par- 
allelogram as  flat  as  OF1FFOJ  and  from  the  combination  of 
vectors  of  the  relative  magnitudes  1 :  6  :100. 

Hence  the  importance  of  the  graphical  method  consists 


34 


ALTERNATING-CURRENT  PHENOMENA.   [§§  24,  25 


not  so  much  in  its  usefulness  for  practical  calculation,  as  to 
aid  in  the  simple  understanding  of  the  phenomena  involved. 

24.  Sometimes  we  can  calqulate  the  numerical  values 
trigonometrically  by  means  of  the  diagram.  Usually,  how- 
ever, this  becomes  too  complicated,  as  will  be  seen  by  trying. 


Fig.  21. 


to  calculate,  from  the  above  transformer  diagram,  the  ratio 
of  transformation.  The  primary  M.M.F.  is  given  by  the 
equation  :  —  • 


2 


sn  «!  , 

an  expression  not  well  suited  as  a  starting-point  for  further 
calculation. 

A  method  is  therefore  desirable  which  combines  the 
exactness  of  analytical  calculation  with  the  clearness  of 
the  graphical  representation. 


o      a 

Fig.  22. 


25.  We  have  seen  that  the  alternating  sine  wave  is 
represented  in  intensity,  as  well,  as  phase,  by  a  vector,  OI, 
which  is  determined  analytically  by  two  numerical  quanti- 
ties—  the  length,  Of,  or  intensity  ;  and  the  amplitude,  AOIy 
or  phase  w,  of  the  wave,  /. 

Instead  of  denoting  the  vector  which  represents  the 
sine  wave  in  the  polar  diagram  by  the  polar  coordinates, 


§25] 


SYMBOLIC  METHOD. 


35 


/  and  o>,  we  can  represent  it  by  its  rectangular  coordinates,. 
a  and  b  (Fig.  22),  where  — 

a  =  /cos  ui  is  the  horizontal  component, 

b  =  /sin  co  is  the  vertica^  component  of  the  sine  wave. 

This  representation  of  the  sine  wave  by  its  rectangular 
components  is  very  convenient,  in  so  far  as  it  avoids  the 
use  of  trigonometric  functions  in  the  combination  or  reso- 
lution of  sine  waves. 

Since  the  rectangular  components  a  and  b  are  the  hori- 
zontal and  the  vertical  projections  of  the  vector  represent- 
ing the  sine  wave,  and  the  projection  of  the  diagonal  of  a 
parallelogram  is  equal  to  the  sum  of  the  projections  of  its 
sides,  the  combination  of  sine  waves  by  the  parallelogram 


Fig.  23. 

• 

law  is  reduced  to  the  addition,  or  subtraction,  of  their 
rectangular  components.  That  is, 

Sine  waves  are  combined,  or  resolved,  by  adding,  or 
subtracting,  their  rectangular  components. 

For  instance,  if  a  and  b  are  the  rectangular  components 
of  a  sine  wave,  /,  and  d  and  b'  the  components  of  another 
sine  wave,  /'  (Fig.  23),  their  resultant  sine  wave,  I0,  has  the 
rectangular  components  a0  =  (a  +  a'),  and  b0  =  (b  +  b'). 

To  get  from  the  rectangular  components,  a  and  b,  of  a 
sine  wave,  its  total  intensity,  i,  and  phase,  to,  we  may  com- 
bine a  and  b  by  the  parallelogram,  and  derive,  — 


tan 


36  ALTERNATING-CURRENT  PHENOMENA.    [§§26,27 

Hence  we  can  analytically  operate  with  sine  waves,  as 
with  forces  in  mechanics,  by  resolving  them  into  their 
rectangular  components. 

26.  To  distinguish,  however,  the  horizontal  and  the  ver- 
tical components  of  sine  waves,  so  as  not  to  be  confused  in 
lengthier  calculation,  we  may  mark,  for  instance,  the  vertical 
components,  by  a  distinguishing  index,  or  the  addition  of 
an  otherwise  meaningless  symbol,  as  the  letter  /,  and  thus 
represent  the  sine  wave  by  the  expression,  — 


which  now  has  the  meaning,  that  a  is  the  horizontal  and  b 
the  vertical  component  of  the  sine  wave  /;  and  that  both 
components  are  to  be  combined  in  the  resultant  wave  of 

intensity,  —  _ 

*=Va2  +  *2, 

and  of  phase,  tan  w  =  b  /  a. 

Similarly,  a  —jb,  means  a  sine  wave  with  a  as  horizon- 
tal, and  —  b  as  vertical,  components,  etc. 

Obviously,  the  plus  sign  in  the  symbol,  a  -\-  jb,  does  not 
imply  simple  addition,  since  it  connects  heterogeneous  quan- 
'  tities  —  horizontal  and  vertical  components  —  but  implies 
combination  by  the  parallelogram  law. 

For  the  present,  /  is  nothing  but  a  distinguishing  index, 
and  otherwise  free  for  definition  except  that  it  is  not  an 
ordinary  number. 

27.  A  wave  of  equal  intensity,  and  differing  in  phase 
from  the  wave  a  +  jb  by  180°,  or  one-half  period,  is  repre- 
sented in  polar  coordinates  by  a  vector  of  opposite  direction, 
and  denoted  by  the  symbolic  expression,  —  a  —  jb.  Or  — 

Multiplying  the  algebraic  expression,  a  +  jb,  of  a  sine  wave 
by  —1  means  reversing  the  wave,  or  rotating  it  tJirough  180°, 
or  one-half  period. 

A  wave  of  equal  intensity,  but  lagging  90°,  or  one- 
quarter  period,  behind  a  -f  jb,  has  (Fig.  24)  the  horizontal 


§  28]  SYMBOLIC  METHOD.  3T 

•component,  —  b,  and  the  vertical  component,  a,  and  is  rep- 
resented algebraically  by  the  expression,  ja  —  b. 
Multiplying,  however,  a  rj-  jb  by  j,  we  get  :  — 


therefore,  if  we  define  the  heretofore  meaningless  symbol, 
j,  by  the  condition,  — 

/2  =  -  1, 
we  have  — 


hence  :  — 

Multiplying  the  algebraic  expression,  a  -{-jb,  of  a  sine  wave 
by  j  means  rotating  the  wave  through  90°,  or  one-quarter  pe- 
riod ;  that  is,  retarding  the  wave  through  one-quarter  period, 


-6 

Fig.  24. 

Similarly,  — 

Multiplying  by  —  j  means  advancing  the  wave  through 
•one-quarter  period. 

since  /2  =  —  1,  j  =  V^T ; 

that  is,  - 

j  is  the  imaginary  unit,  and  the  sine  wave  is  represented 
by  a  complex  imaginary  quantity,  a  -f  jb. 

As  the  imaginary  unit  j  has  no  numerical  meaning  in 
the  system  of  ordinary  numbers,  this  definition  of/  =  V— 1 
does  not  contradict  its  original  introduction  as  a  distinguish- 
ing index.  For  a  more  exact  definition  of  this  complex 
imaginary  quantity,  reference  may  be  made  to  the  text  books 
of  mathematics. 

28.  In  the  polar  diagram  of  time,  the  sine  wave  is 
represented  in  intensity  as  well  as  phase  by  one  complex 

quantity  — 

1  a  +  jb, 


38  ALTERNATING-CURRENT  PHENOMENA.  [§29 

where  a  is  the  horizontal  and  b  the  vertical  component  of 
the  wave  ;  the  intensity  is  given  by  — 


the  phase  by  — 

tan  w  =  -  , 
a 

and 

a  =  i  cos  o>, 

b  =  i  sin  to  ; 

hence  the  wave  a  -\-jb  can  also  be  expressed  by  — 
i  (cos  W  +y  sin  w)> 

or,  by  substituting  for  cos  w  and  sin  £  their  exponential 

expressions,  we  obtain  — 

ie^. 

Since  we  have  seen  that  sine  waves  may  be  combined 
or  resolved  by  adding  or  subtracting  their  rectangular  com- 
ponents, consequently  :  — 

Sine  waves  may  be  combined  or  resolved  by  adding  or 
subtracting  their  complex  algebraic  expressions. 

For  instance,  the  sine  waves,  — 


and 

combined  give  the  sine  wave  — 

f=(a  +  a')+j(b  +  bf). 

It  will  thus  be  seen  that  the  combination  of  sine  waves 
is  reduced  to  the  elementary  algebra  of  complex  quantities. 

29.  If  /=  i  -\-ji'  is  a  sine  wave  of  alternating  current, 
and  r  is  the  resistance,  the  E.M.F.  consumed  by  the  re- 
sistance is  in  phase  with  the  current,  and  equal  to  the  prod- 
uct of  the  current  and  resistance.  Or  — 

rl  =  ri  -\-  jri'  . 

If  L  is  the  inductance,  and  x  —  2  tr  NL  the  reactance, 
the  E.M.F.  produced  by  the  reactance,  or  the  counter 


§  29]  SYMBOLIC  METHOD.  39 

E.M.F.   of    self-inductance,  is  the  product  of   the    current 
and   reactance,   and    lags    90°    behind   the    current;    it    is, 

therefore,  represented  by  the  expression  — 

j 

jxl  =jit?ci  —  xir. 

The  E.M.F.  required  to  overcome  the  reactance  is  con- 
sequently 90°  ahead  of  the  current  (or,  as  usually  expressed, 
the  current  lags  90°  behind  the  E.M.F.),  and  represented 
by  the  expression  — 

—jxj~=  —  jxi  +  xi'. 

Hence,  the  E.M.F.  required  to  overcome  the  resistance, 
r,  and  the  reactance,  x,  is  — 

(r-jx)I- 
that  is  — 

Z  =  r  —  jx  is  the  expression  of  the  impedance  of  the  cir- 
cuit, in  complex  quantities. 

Hence,  if  /  =  i  -\-ji'  is  the  current,  the  E.M.F.  required 
to  overcome  the  impedance,  Z  =  r  —  jx,  is  — 


hence,  since  j2  =  —  1 

E  =  (ri  +  xi')  +  j  (rir  -  xi)  ; 

or,  if  E  =  e  +je'  is  the  impressed  E.M.F.,  and  Z  =  r  —  jx 
the  impedance,  the  current  flowing  through  the  circuit  is  :  — 


or,  multiplying  numerator  and  denominator  by  (r  -\-jx)  to 
eliminate  the  imaginary  from  the  denominator,  we  have  — 


(e  +X)  (r  +  •/*)       er—Jx  _._   .Jr  +  ex 


T  = 

or,  if  E  =  e  -\-je'  is  the  impressed  E.M.F.,  and  /  =  /  -f-  jir 
the  current  flowing  in  the  circuit,  its  impedance  is  — 

z  =  £_  = 

' 


|*  4-  /' 


40  AL  TERN  A  TING-CURRENT  PHENOMENA.  [§§  30,  3  1 

30.    If  C  is  the  capacity  of   a  condenser  in    series    in 
a  circuit  of  current  /  =  i  +  /*',  the  E.M.F.  impressed  upon 

the  terminals  of  the  condenser  is  E  =  -  -  ,  90°  behind 


the  current;  and  may  be  represented  by  —  3-  -  ,  or  jx^  /, 

—  7T  2.  V  Cx 

where  x^  =  -  -  —  -  is  the  capacity  reactance  or  condensance 

—  j 


of  the  condenser. 

Capacity  reactance  is  of  opposite  sign  to  magnetic  re- 
actance ;  both  may  be  combined  in  the  name  reactance. 

We  therefore  have  the  conclusion  that 

If  r  =  resistance  and  L  =  inductance, 

then  x  =  2  irNL  =  magnetic  reactance. 

If  C  =  capacity,  Xi  =  -  =  capacity  reactance,  or  conden- 

2i  TT  yvCx 

sance  ; 

Z  =  r  —  j  (x  —  Xi)j  is  the  impedance  of  the  circuit. 

Ohm's  law  is  then  reestablished  as  follows  : 


The  more  general  form  gives  not  only  the  intensity  of 
the  wave,  but  also  its  phase,  as  expressed  in  complex 
quantities. 

31.  Since  the  combination  of  sine  waves  takes  place  by 
the  addition  of  their  symbolic  expressions,  Kirchhoff  s  laws 
are  now  reestablished  in  their  original  form  :  — 

a.)  The  sum  of  all  the  E.M.Fs.  acting  in  a  closed  cir- 
cuit equals  zero,  if  they  are  expressed  by  complex  quanti- 
ties, and  if  the  resistance  and  reactance  E.M.Fs.  are  also 
considered  as  counter  E.M.Fs. 

b.)  The  sum  of  all  the  currents  flowing  towards  a  dis- 
tributing point  is  zero,  if  the  currents  are  expressed  as 
complex  quantities. 


§32]  SYMBOLIC  METHOD.  41 

Since,  if  a  complex  quantity  equals  zero,  the  real  part  as- 
well  as  the  imaginary  part  must  be  zero  individually,  if 

a  +jb  =  0,      .    a  =  0,  b  =  0, 

and  if  both  the  E.M.Fs.  amd  currents  are  resolved,  we 
find:  — 

a.)  The  sum  of  the  components,  in  any  direction,  of  all 
the  E.M.Fs.  in  a  closed  circuit,  equals  zero,  if  the  resis- 
tance and  reactance  are  considered  as  counter  E.M.Fs. 

b.)  The  sum  of  the  components,  in  any  direction,  of  all 
the  currents  flowing  towards  a  distributing  point,  equals 
zero. 

Joule's  Law  and  the  energy  equation  do  not  give  a 
simple  expression  in  complex  quantities,  since  the  effect  or 
power  is  a  quantity  of  double  the  frequency  of  the  current 
or  E.M.F.  wave,  and  therefore  cannot  be  represented  as 
a  vector  in  the  diagram. 

In  what  follows,  complex  quantities  will  always  be  de- 
noted by  capitals,  absolute  quantities  and  real  quantities  by 
small  letters. 

32.  Referring  to  the  instance  given  in  the  fourth 
chapter,  of  a  circuit  supplied  with  an  E.M.F.,  E,  and  a  cur- 
rent, /,  over  an  inductive  line,  we  can  now  represent  the 
impedance  of  the  line  by  Z  =  r  — jx,  where  r  =  resistance, 
x  =  reactance  of  the  line,  and  have  thus  as  the  E.M.F. 
at  the  beginning  of  the  line,  or  at  the  generator,  the 

expression  — 

E0  =  E  +  ZL 

Assuming  now  again  the  current  as  the  zero  line,  that 
is,  /  =  2,  we  have  in  general  — 

E0  =  E  +  ir-jix-, 

hence,  with  non-inductive  load,  or  E  =  e, 
EO  =  («  +  t'r)  —fix, 


e0   =  V(<?  +  try  +  (t'x)*,     tan  w0  =  — 


tx 
ir 


42  ALTERNATING-CURRENT  PHENOMENA.  [§32 

In  a    circuit  with    lagging  current,  that    is,  with    leading 
E.M.F.,  E  =  e  —je',  and 

£0=  e  —  jJ  +  (r  —  jx)  i 


or  e0  =        *+z+       +  *,     tan  w0  = 


e  -\-  ir 

In  a    circuit  with  leading    current,  that    is,    with    lagging 
E.M.F.,  E  =  e  +je',  and 


e0  =  -(e  +  ir*  +  (    -  /•*)*,     tan  Z>0  = 
values  which  easily  permit  calculation. 


e  —  ix 


•§33]  TOPOGRAPHIC  METHOD.  43 


« 
i 


-  CHAPTER    VI. 

TOPOGRAPHIC    METHOD. 

33.  In  the  representation  of  alternating  sine  waves  by 
vectors  in  a  polar  diagram,  a  certain  ambiguity  exists,  in  so 
far  as  one  and  the  same  quantity  —  an  E.M.F.,  for  in- 
stance —  can  be  represented  by  two  vectors  of  opposite 
direction,  according  as  to  whether  the  E.M.F.  is  considered 
as  a  part  of  the  impressed  E.M.F.,  or  as  a  counter  E.M.F. 
.This  is  analogous  to  the  distinction  between  action  and 
reaction  in  mechanics. 


Fig.  25. 


Further,  it  is  obvious  that  if  in  the  circuit  of  a  gener- 
ator, G  (Fig.  25),  the  current  flowing  from  terminal  A  over 
resistance  R  to  terminal  B,  is  represented  by  a  vector  OT 
(Fig.  26),  or  by  /  =  i  -\-ji',  the  same  current  can  be  con- 
sidered as  flowing  in  the  opposite  direction,  from  terminal 
B  to  terminal  A  in  opposite  phase,  and  therefore  represented 
by  a  vector  OI^  (Fig.  26),  or  by  /j  =  —  i  —ji'. 

Or,  if  the  difference  of  potential  from  terminal  B  to 
terminal  A  is  denoted  by  the  E  =  e  +  je' ,  the  difference 
of  potential  from  A  to  B  is  El  =  —  e  —je'. 


44 


ALTERNATING-CURRENT  PHENOMENA. 


[§34 


Hence,  in  dealing  with  alternating-current  sine  waves, 
it  is  necessary  to  consider  them  in  their  proper  direction 
with  regard  to  the  circuit.  Especially  in  more  complicated 
circuits,  as  interlinked  polyphase  systems,  careful  attention 
has  to  be  paid  to  this  point. 


Fig.  28. 


34.  Let,  for  instance,  in  Fig.  27,  an  interlinked  three- 
phase  system  be  represented  diagrammatically,  and  consist- 
ing of  three  E.M.Fs.,  of  equal  intensity,  differing  in  phase 
by  one-third  of  a  period.  Let  the  E.M.Fs.  in  the  direction 


Fig.  27. 


from  the  common  connection  O  of  the  three  branch  circuits 
to  the  terminals  Alt  A2,  As,  be  represented  by  Ely  E2,  EB. 
Then  the  difference  of  potential  from  A2  to  A1  is  E%  —  Ely 


since  the  two  E.M.Fs.,  E    and 


are  connected  in  cir- 


cuit between  the  terminals  A*   and  A*,  in  the  direction,. 


§34]  TOPOGRAPHIC  METHOD.  45 

A1  —  O  —  A2;  that  is,  the  one,  ^2,  in  the  direction  OAZ, 
from  the  common  connection  to  terminal,  the  other,  Elt  in 
the  opposite  direction,  A^O,  .from  the  terminal  to  common 
connection,  and  represented  by  —  El.  Conversely,  the  dif- 
ference of  potential  from  A± To  A2  is  E^  —  E2. 

It  is  then  convenient  to  go  still  a  step  farther,  and 
drop,  in  the  diagrammatic  representation,  the  vector  line 
altogether ;  that  is,  denote  the  sine  wave  by  a  point  only, 
the  end  of  the  corresponding  vector. 

Looking  at  this  from  a  different  point  of  view,  it  means 
that  we  choose  one  point  of  the  system  —  for  instance,  the 
common  connection  O  —  as  a  zero  point,  or  point  of  zero 
potential,  and  represent  the  potentials  of  all  the  other  points 
of  the  circuit  by  points  in  the  diagram,  such  that  their  dis- 
tances from  the  zero  point  gives  the  intensity ;  their  ampli- 
tude the  phase  of  the  difference  of  potential  of  the  respective 
point  with  regard  to  the  zero  point ;  and  their  distance  and 
amplitude  with  regard  to  other  points  of  the  diagram,  their 
difference  of  potential  from  these  points  in  intensity  and 
phase. 


E? 

Fig.  28. 


Thus,  for  example,  in  an  interlinked  three-phase  system 
with  three  E.M.Fs.  of  equal  intensity,  and  differing  in  phase 
by  one-third  of  a  period,  we  may  choose  the  common  con- 
nection of  the  star-connected  generator  as  the  zero  point, 
and  represent,  in  Fig.  28,  one  of  the  E.M.Fs.,  or  the  poten- 


46 


ALTERNATING-CURRENT  PHENOMENA. 


[§35 


tial  at  one  of  the  three-phase  terminals,  by  point  E^.  The 
potentials  at  the  two  other  terminals  will  then  be  given  by 
the  points  E2  and  Ez,  which  have  the  same  distance  from 
O  as  Elt  and  are  equidistant  from  E1  and  from  each  other. 


Fig.  29. 


The  difference  of  potential  between  any  pair  of  termi- 
nals —  for  instance  E1  and  E^  —  is  then  the  distance  E^E^  , 
Z,  according  to  the  direction  considered. 


or 


I.  E° 


35.  If,  now,  in  Fig.  29,  a  current,  Ilt  in  phase  with 
E.M.F.,  Elt  passes  through  a  circuit,  the  counter  E.M.F. 
of  resistance,  r,  is  Er  =  fr,  in  opposition  to  /j  or  E19 


§  35] 


TOPOGRAPHIC  METHOD. 


47 


.and  hence  represented  in  the  diagram  by  point  Er,  and 
its  combination  with  E^  by  E±.  The  counter  E.M.F.  of 
reactance,  x,  is  Ex  =  Ix,  90°  behind  the  current  flt  or 
E.M.F.,  Elf  and  therefore  represented  by  point  Ex,  and 
giving,  by  its  combination  with  E^,  the  terminal  potential 
of  the  generator  E^,  which,  as  seen,  is  less  than  the 
E.M.F.,  Elf 

If  all  the  three  branches  are  loaded  equally  by  three 
•currents  flowing  into   a   non-inductive  circuit,  and  thus  in 


Fig.  31. 


phase  with  the  E.M.Fs.  at  the  generator  terminals  (repre- 
sented in  the  diagram,  Fig.  30,  by  the  points  Elt  Ez,  EZJ 
equidistant  from  each  other,  and  equidistant  from  the  zero 
point,  O),  the  counter  E.M.Fs.  of  resistance,  Irt  are  repre- 
sented by  the  distances  EE1 ',  as  E-^E^  etc.,  in  phase  with 
the  currents,  /;  and  the  counter  E.M.Fs.  of  reactance,  fx, 
are  represented  by  the  distance,  E1 E°  in  quadrature  with 
the  current,  thereby  giving,  at  the  generator  E.M.Fs.,  the 
points  E{,  Ez°,  E£. 

Thus,  the  triangle  of  generator  E.M.Fs.  EfE^Ej,  pro- 
duces, with    equal  load   on    the  three   branches   and  non- 


48  ALTERNATING-CURRENT  PHENOMENA.  [§36 

inductive  circuit,  the  equilateral  triangle,  E1E^EZ9  of  ter- 
minal potentials. 

If  the  load  is  inductive,  and  the  currents,  /,  lag  behind 
the  terminal  voltages,  E,  by,  say,  40°,  we  get  the  diagram 
shown  in  Fig.  31,  which  explains  itself,  and  shows  that  the 
drop  of  potential  in  the  generator  is  larger  on  an  inductive 
load  than  on  a  non-inductive  load. 

Conversely,  if  the  currents  lead  the  terminal  E.M.Fs. 
by,  say,  40°,  as  shown  in  Fig.  32,  the  drop  of  potential  in 
the  generator  is  less,  or  a  rise  may  even  take  place. 


36.  If,  however,  only  one  branch  of  the  three-phase 
circuit  is  loaded,  as,  for  instance,  E^EJ,  the  E.M.F.  pro- 
ducing the  current  (Fig.  33),  is  E^E^  ;  and,  if  the  current 
lags  20°,  it  has  the  direction  Of,  where  Of  forms  with 
E^EI  the  angle  eo  —  20°  ;  that  is,  the  current  in  E^  is  Oflt 
and  the  current  in  E^  is  6>/2 ,  the  return  current  of  OIV 

Hence  the  potential  at  the  first  terminal  is  E^,  as  de- 
rived by  combining  with  Ef  the  resistance  E.M.F.,  E^E^9 
in  phase,  and  the  reactance,  E.M.F.,  E±E19  in  quadrature, 
with  the  current ;  and  in  the  same  way,  the  E.M.F.  at  the 


§36]  TOPOGRAPHIC  METHOD.  49 

second  terminal  is  -ZT2,  derived  by  the  combination  of  E<£ 
with  E<£E^  in  phase,  and  E^E^  in  quadrature,  with  the 
current. 

Hence  the  three  terminal,  potentials  are  now,  Elt  E^ 
EB°,  and  the  differences  of  potential  between  the  terminals 
of  the  generators  are  the  sides  of  the  triangle,  E^E^E^ ; 
or,  in  other  words,  the  equilateral  triangle  of  E.M.F., 
E^E^E^y  produces  at  the  generator  terminals  the  triangle 
of  voltages,  E-^E^EJ,  whose  three  sides  are  unequal ;  one, 
E^E^  or  the  loaded  branch,  being  less  than  E^EJ,  or  the 
two  unloaded  branches.  That  is,  the  one  has  decreased, 
the  other  has  increased,  and  the  system  has  become  un- 
balanced- 


Fig.  33. 

In  the  same  manner,  if  two  branches,  EfE£,  and 
E£E£,  are  loaded,  and  the  third,  E^E^,  is  unloaded,  and 
the  currents  lag  20°,  we  find  the  current  73  in  Ef  to  be 
20°  behind  EfEf,  the  current  7j  in  Ez°  20°  behind  EfEf, 
and  the  current  72  in  E<£,  the  common  return  of  II  and 
73 ;  by  combining  again  the  generator  E.M.Fs.,  E°,  with 
the  resistance  E.M.Fs.,  E°E',  in  phase,  and  the  reactance 
E.M.Fs.,  E' E,  in  quadrature,  with  the  respective  currents, 
we  get  the  terminal  potentials,  E.  We  thus  see  that  the 
E.M.F.  triangle,  E^EjEj,  is,  by  loading  two  branches, 
changed  to  an  unbalanced  triangle  of  terminal  voltages, 
E-^E^E^,  as  shown  in  Fig.  34. 


ALTERNATING-CURRENT  PHENOMENA. 


[§  37 


If  all  the  three  branches  of  the  three-phase  system  are 
loaded  equally,  we  see,  from  Fig.  31,  that  the  system 
remains  balanced. 

37.  As  another  instance,  we  may  consider  the  unbal- 
ancing of  a  two-phase  system  with  a  common  return. 

If,  in  a  two-phase  system,  we  choose  the  potential  of  the 
common  return  at  the  generator  terminal  as  zero,  the  poten- 
tials of  the  two  outside  terminals  of  the  generator  are  repre- 


sented  by  E£  and  E2°,  at  right  angles  to  each  other,  and 
equidistant  from  O,  as  shown  in  Fig.  35. 

Let,  now,  both  branches  be  loaded  equally  by  currents 
lagging  40°.  Then,  the  currents  in  Ef  and  E£  are  repre- 
sented by  7j  and  72,  and  their  common  return  current  by 
73.  If,  now,  these  currents  are  sent  over  lines  containing 
resistance  and  reactance,  we  get  the  potentials  at  the  end 
of  the  line  by  combining  the  generator  potentials  E^,  E2° 
and  O,  with  the  resistance  E.M.Fs.,  E^E^  E20E%  and 
OE%,  in  phase  with  the  currents,  and  with  the  reactance 
E.M.Fs.,  E^Elt  E^E2,E^EB,  in  quadrature  with  the  cur- 
rents ;  and  thereby  derive  as  the  potentials  at  the  end  of  the 
line  the  points  E19  E2,  E3,  which  form  neither  an  isosceles 


§  37]  TOPOGRAPHIC  METHOD.  51 

nor  a  rectangular  triangle.  That  is,  the  two-phase  system 
with  a  common  return  becomes,  even  at  equal  distribution 
of  load,  unbalanced  in  intensity  and  in  phase. 

These  instances  will  be*  sufficient  to  explain  the  general 
method  of  topographic  refftesentation'  of  alternating  sine 
waves. 

It  is  obvious  now,  since  the  potential  of  every  point  of 


Fig.  35. 

the  circuit  is  represented  by  a  point  in  the  topographic 
diagram,  that  the  whole  circuit  will  be  represented  by  a 
closed  figure,  which  may  be  called  the  topographic  circuit 
characteristic. 

Such  a  characteristic  is,  for  instance,  OE^E^E^Ef^E^ 
in  Figs.  31  to  34,  etc.  ;  further  instances  are  shown  in  the 
following  chapters,  as  curved  characteristics  in  the  chapter 
on  distributed  capacity,  etc. 


52  ALTERNATING-CURRENT  PHENOMENA.  [§38 


CHAPTER    VII. 

ADMITTANCE,    CONDUCTANCE,    SUSCEPTANCE. 

38.  If  in  a  continuous-current  circuit,  a  number  of 
resistances,  rlt  r2,  rs,  .  .  .  are  connected  in  series,  their 
joint  resistance,  R,  is  the  sum  of  the  individual  resistances 
R  =  r^  +  r^  +  rz  +  .  .  . 

If,  however,  a  number  of  resistances  are  connected  in 
multiple  or  in  parallel,  their  joint  resistance,  R,  cannot 
be  expressed  in  a  simple  form,  but  is  represented  by  the 
expression  :  — 


Hence,  in  the  latter  case  it  is  preferable  to  introduce,  in- 
stead of  the  term  resistance,  its  reciprocal,  or  inverse  value, 
the  term  conductance,  g  =  1  /  r.  If,  then,  a  number  of  con- 
ductances, glt  g^y  gz,  .  .  .  are  connected  in  parallel,  their 
joint  conductance  is  the'sum  of  the  individual  conductances, 
or  G  =  g^  +  jr2  -f  gz  -(-  .  .  .  When  using  the  term  con- 
ductance, the  joint  conductance  of  a  number  of  series- 
connected  conductances  becomes  similarly  a  complicated 
expression  — 

G=  _  -  _  . 

1  +  1  +  ^  +  ... 

gi       g*       g* 

Hence  the  term  resistance  is  preferable  in  case  of  series 
connection,  and  the  use  of  the  reciprocal  term  conductance 
in  parallel  connections  ;  therefore, 

The  joint  resistance  of  a  number  of  series-connected  resis- 
tances is  eqtial  to  the  sum  of  the  individual  resistances  ;  the 


§39]      ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE.          53 

joint  conductance  of  a  number  of  parallel-connected  conduc- 
tances is  equal  to  the  sum  of  tJie  individual  conductances. 

39.  In  alternating-cunrent  circuits,  instead  of  the  term 
resistance  we  have  the  termMmpedance,  Z  =  r  — jx,  with  its 
two  components,  the  resistance,  r,  and  the  reactance,  x,  in  the 
formula  of  Ohm's  law,  E  =  IZ.  The  resistance,  r,  gives 
the  component  of  E.M.F.  in  phase  with  the  current,  or  the 
energy  component  of  the  E.M.F.,  Ir\  the  reactance,  x, 
gives  the  component  of  the  E.M.F.  in  quadrature  with  the 
current,  or  the  wattless  component  of  E.M.F.,  Ix ;  both 
combined  give  the  total  E.M.F., — 


Since  E.M.Fs.  are  combined  by  adding  their  complex  ex- 
pressions, we. have  : 

The  joint  impedance  of  a  number  of  series-connected  impe- 
dances is  the  sum  of  the  individual  impedances,  when  expressed 
in  complex  quantities. 

In  graphical  representation  impedances  have  not  to  be 
added,  but  are  combined  in  their  proper  phase  by  the  law 
of  parallelogram  in  the  same  manner  as  the  E.M.Fs.  corre- 
sponding to  them.  The  term  impedance  becomes  incon-- 
venient,  however,  when  dealing  with  parallel-connected 
circuits  ;  or,  in  other  words,  when  several  currents  are  pro- 
duced by  the  same  E.M.F.,  such  as  in  cases  where  Ohm's 
law  is  expressed  in  the  form, 

T-E 
~~Z' 

It  is  preferable,  then,  to  introduce  the  reciprocal  of 
Impedance,  which  may  be  called  the  admittance  of  the 

circuit,  or 

v _ 

~  Z  '    . 

As  the  reciprocal  of  the  complex  quantity,  Z  =  r  —  jx,  the 
admittance  is  a  complex  quantity  also,  or 


54  ALTERNATING-CURRENT  PHENOMENA.  [§  4O 

it  consists  of  the  component  g,  which  represents  the  co- 
efficient of  current  in  phase  with  the  E.M.F.,  or  energy 
current,  gEt  in  the  equation  of  Ohm's  law,  — 

I  =YE  =  (g  +  jb)  E, 

and  the  component  b,  which  represents  the  coefficient  of 
current  in  quadrature  with  the  E.M.F.,  or  wattless  com- 
ponent of  current,  bE. 

g  may  be  called  the  conductance,  and  b  the  susceptance,. 
of  the  circuit.  Hence  the  conductance,  gy  is  the  energy 
component,  and  the  susceptance,  b,  the  wattless  component, 
of  the  admittance,  Y  =  g  +  jb,  while  the  numerical  value  of 

admittance  is  — 

y  =  V?3  +  ^ ; 

the  resistance,  r,  is  the  energy  component,  and  the  reactance,. 
x,  the  wattless  component,  of  the  impedance,  Z  =  r  —  jx,. 
the  numerical  value  of  impedance  being  — 

z  =  V/-*  +  x2. 

40.  As  shown,  the  term  admittance  implies  resolving 
the  current  into  two  components,  in  phase  and  in  quadra- 
ture with  the  E.M.F.,  or  the  energy  current  and  the  watt- 
less current  ;  while  the  term  impedance  implies  resolving 
the  E.M.F.  into  two  components,  in  phase  and  in  quad- 
rature with  the  current,  or  the  energy  E.M.F.  and  the 
wattless  E.M.F. 

It  must  be  understood,  however,  that  the  conductance 
is  not  the  reciprocal  of  the  resistance,  but  depends  upon 
the  resistance  as  well  as  upon  the  reactance.  Only  when  the 
reactance  x  ==  0,  or  in  continuous-current  circuits,  is  the 
conductance  the  reciprocal  of  resistance. 

Again,  only  in  circuits  with  zero  resistance  (r  =  0)  is 
the  susceptance  the  reciprocal  of  reactance ;  otherwise,  the 
susceptance  depends  upon  reactance  and  upon  resistance. 

The  conductance  is  zero  for  two  values  of  the  resistance  :  — 

1.)  If  r  —  oo  ,  or  x  =  oo  ,  since  in  this  case  no  current 
passes,  and  either  component  of  the  current  =  0. 


§  4O]      ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE.          55 

2.)  If  r  —  0,  since  in  this  case  the  current  which  passes 
through  the  circuit  is  in  quadrature  with  the  E.M.F.,  and 
thus  has  no  energy  component. 

Similarly,  the  susceptance,  b,  is  zero  for  two  values  of 
the  reactance  :  — 

1.)    If  x  =  oo  ,  or  r  =  GO  . 

2.)    If  x  =  0. 

From  the  definition  of  admittance,  Y  —  g -\- jb,  as  the 
reciprocal  of  the  impedance,  Z  =  r  — jx, 

we  have  Y  =  -^  ,  or,  g  +  jb  =  -^—  • 

Z  r—jx 

or,  multiplying  numerator  and  denominator  on  the  right  side 
by  (r 


hence,  since 

(r  —  jx)  (r  +  jx)  =  r2  -f  x2  = 


or  8  =  ~TT^  =  ^2 ' 


r*  +  x2 
and  conversely 


By  these  equations,  the  conductance  and  susceptance  can- 
be  calculated  from  resistance  and  reactance,  and  conversely.. 
Multiplying  the  equations  for  g  and  r>  we  get  :  — 


hence,  *2/  =  (^  +  ^2)  (g2  +  £2)  =  1 ; 

1  1  )  the  absolute  value  of 


and, 


y       yV2  +  b2 '     i  impedance  ; 

)  the   absolute  value  of. 
admittance. 


ALTERNATING-CURRENT  PHENOMENA. 


[§41 


41.  If,  in  a  circuit,  the  reactance,  x,  is  constant,  and  the 
resistance,  r,  is  varied  from  r  —  0  to  r  =  oo  ,  the  susceptance, 
b,  decreases  from  £  =  1  /  .r  at  r  =  0,  to  $  =  0  at  r  =  oo  ; 
while  the  conductance,  g  =  0  at  r  =  0,  increases,  reaches 
a  maximum  for  r  =  x,  where  g  =  1  /  2  r  is  equal  to  the 
susceptance,  or  g  =  b,  and  then  decreases  again,  reaching 
•  =  0  at  r  =  oo  . 


OHN 

2.0 
1.9 
1.8 
1.7 
1.6 
1.5 
1.4 
1.3 
1.2 
1.1 
1.0 
.9 
.8 
.7 

s 

^ 

\ 

V 

\ 

RE/ 

crt 

NC 

CO 

NS1 

AN! 

=  .l 

OH 

MS 

/ 

\ 

\ 

s 

\ 

\ 

/ 

\ 

v 

\ 

/ 
S 

\ 

\ 

/ 
/ 

\ 

1 

'X 

/ 

> 

\ 

X 

^ 

•P 

\ 

§ 

\ 

,# 

<<," 

\ 

^ 

\/ 

/ 

/ 

r 

\ 

•^>; 

'XN 

X 

/ 

\ 

/ 

' 

^ 

\, 

X 

7 

/ 

i 

t> 

^\ 

"^x 

\ 

/ 

s 

^•^ 

°j 
< 

^N 

"**** 

^ 

^ 

^^^ 

r, 

3 

r^ 

^ 

\ 

•-^, 

C^; 

^ 

.4 

o 

/? 

^2 

X 

I 

^r 

^ 

x 

^ 

o 

i 

3 

^ 

^ 

.1 

0 
•    ( 

8 

""*^^. 

~  — 

—  ~- 

RESISJAN 

DE: 

%  O^MS 

1         1 

i  - 

>     .1     ~.       3     4      .5     .Q      .7     .8      .9     1.0   1.1  1.2    1.3    1.1  1.5    1.0   1.7   1.8 

Fig.  36. 

In  Fig.  36,  for  constant  reactance  x  =  .5  ohm,  the  vari- 
ation of  the  conductance,  g,  and  of  the  susceptance,  b,  are 
shown  as  functions  of  the  varying  resistance,  r.  As  shown, 
the  absolute  value  of  admittance,  susceptance,  and  conduc- 
tance are  plotted  in  full  lines,  and  in  dotted  line  the  abso- 
lute value  of  impedance, 


§41]      ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE.         57 

Obviously,  if  the  resistance,  r,  is  constant,  and  the  reac- 
tance, x,  is  varied,  the  values  of  conductance  and  susceptance 
are  merely  exchanged,  the  conductance  decreasing  steadily 
from  g  —  1  /  r  to  0,  and  the  susceptance  passing  from  0  at 
x  =  0  to  the  maximum,  b  =4  I  ^  r  —  g  =\  /  ^x  at  x  =  r> 
and  to  b  —  0  at  x  =  oo  . 

The  resistance,  r,  and  the  reactance,  x,  vary  as  functions 
of  the  conductance,  g,  and  the  susceptance,  b,  varies,  simi- 
larly to  g  and  b,  as  functions  of  r  and  x. 

The  sign  in  the  complex  expression  of  admittance  is 
always  opposite  to  that  of  impedance  ;.it  follows  that  if  the 
current  lags  behind  the  E.M.F.,  the  E.M.F.  leads  the  cur- 
rent, and  conversely. 

We  can  thus  express  Ohm's  law  in  the  two  forms  — 

E  =  IZ, 
I  =  EY, 
and  therefore  — 

The  joint  impedance  of  a  number  of  series-connected  im- 
pedances is  equal  to  the  sttm  of  the  individual  impedances  ; 
the  joint  admittance  of  a  number  of  parallel-connected  admit- 
tances, if  expressed  in  complex  quantities,  is  equal  to  the  sum 
of  the  individual  admittances.  In  diagrammatic  represen- 
tation, combination  by  the  parallelogram  law  takes  the  place 
of  addition  of  the  complex  quantities. 


.58  ALTERNATING-CURRENT  PHENOMENA.     [§§42,43 


CHAPTER   VIII. 

CIRCUITS    CONTAINING    RESISTANCE,    INDUCTANCE,    AND 
CAPACITY. 

42.  Having,  in  the  foregoing,  reestablished  Ohm's  law 
and  Kirchhoff s   laws  as  being  also  the  fundamental  laws 
of  alternating-current  circuits,  or,  as  expressed  in  their  com- 
plex form,  * 

E  =  ZI,  or,  /  =  YE, 

.and  ^E  =  0  in  a  closed  circuit, 

27  =  0  at  a  distributing  point, 

where  Ey  7,  Z,  Y,  are  the  expressions  of  E.M.F.,  current, 
Impedance,  and  admittance  in  complex  quantities,  —  these 
laws  representing  not  only  the  intensity,  but  also  the  phase, 
of  the  alternating  wave,  —  we  can  now  —  by  application  of 
these  laws,  and  in  the  same  manner  as  with  continuous- 
current  circuits,  keeping  in  mind,  however,  that  E,  7,  Zy  Y, 
are  complex  quantities  —  calculate  alternating-current  cir- 
cuits and  networks  of  circuits  containing  resistance,  induc- 
tance, and  capacity  in  any  combination,  without  meeting 
with  greater  difficulties  than  when  dealing  with  continuous- 
current  circuits. 

It  is  obviously  not  possible  to  discuss  with  any  com- 
pleteness all  the  infinite  varieties  of  combinations  of  resis- 
tance, inductance,  and  capacity  which  can  be  imagined,  and 
which  may  exist,  in  a  system  or  network  of  circuits  ;  there- 
fore only  some  of  the  more  common  combinations  will  here 
be  considered. 

1.)    Resistance  in  series  with  a  circuit. 

43.  In    a    constant-potential     system    with    impressed 
E.M.F., 


§43]  RESISTANCE,  INDUCTANCE,  CAPACITY.  59 

let  the  receiving  circuit  of  impedance 


Z  =  r  —  jxj  z  =  -\/r'2  -f-  x'2> 

be  connected  in  series  with  a  resistance,  r0  . 
The  total  impedance  of  tMe  circuit  is  then 

Z  +  r0  =  r  +  r0—jx; 
hence  the  current  is 


Z  +  r0       r+r0-jx          (r  +  r0)*  -f  x*  ' 
.and  the  E.M.F.  of  the  receiving  circuit,  becomes 
E  =  IZ  =  E°  (r  —• 


r  +  r0  -jx  (r  +  r0)2  +  x* 

=  EQ{#  +  rr0  —jr0x}   t 
z*  +  2  rr0  +  r02 

or,  in  absolute  values  we  have  the  following  :  — 
Impressed  E.M.F., 


current, 


V^T^y2!^2       V*2  +  2  rr0  +  r*  ' 
E.M.F.  at  terminals  of  receiver  circuit, 


'  o)    '  V  z   -f-  4  rrc 

difference  of  phase  in  receiver  circuit, 

tan  c3  =  -  ; 

x 

difference  of  phase  in  supply  circuit, 
tan  wn  =  r*~r° 


x 

a.}    If  x  is  negligible  with  respect  to  r,  as  in  a  non-induc- 
tive receiving  circuit, 

T  —         Eo  77   _.     TT  r 


and  the  current  and  E.M.F.  at  receiver  terminals  decrease 
.steadily  with  increasing  r0 . 


60  ALTERNATING-CURRENT  PHENOMENA.  [§44 

b.)    If  r  is  negligible  compared  with  x,  as  in  a  wattless 
receiver  circuit, 

T—       E° 


or,  for  small  values  of  r0  , 


that  is,  the  current  and  E.M.F.  at  receiver  terminals  remain 
approximately  constant  for  small  values  of  r0,  and  then  de- 
crease  with  increasing  rapidity. 

44.  In  the  general  equations,  x  appears  in  the  expres- 
sions for  /  and  E  only  as  .r2,  so  that  /  and  E  assume  the 
same  value  when  x  is  negative,  as  when  x  is  positive  ;  or,  in 
other  words,  series  resistance  acts  upon  a  circuit  with  leading 
current,  or  in  a  condenser  circuit,  in  the  same  way  as  upon  a 
circuit  with  lagging  current,  or  an  inductive  circuit. 

For  a  given  impedance,  #,  of  the  receiver  circuit,  the  cur- 
rent /,  and  E.M.F.,  E,  are  smaller,  as  r  is  larger  ;  that  is, 
the  less  the  difference  of  phase  in  the  receiver  circuit. 

As  an  instance,  in  Fig.  37  are  shown  in  dotted  lines  the 
current,  /,  and  the  E.M.F.,  E,  at  the  receiver  circuit,  for 
E0  =  const.  =  100  volts,  z  =  1  ohm,  and"- 

a.)    r0  =  .2  ohm         (Curve    I.) 
£.)    r0  =  .8  ohm         (Curve  II.) 

with  values  of  reactance,  x=  V^2  —  r2,  for  abscissae,  from 
x  =  +  1.0  to  x  =  +  1.0  ohm. 

As  shown,  /  and  E  are  smallest  for  x  —  0,  r  =  1.0, 
or  for  the  non-inductive  receiver  circuit,  and  largest  for 
x  =  zt  1.0,  r  =  0,  or  for  the  wattless  circuit,  in  which  latter 
a  series  resistance  causes  but  a  very  small  drop  of  potential. 

Hence  the  control  of  a  circuit  by  series  resistance  de- 
pends upon  the  difference  of  phase  in  the  circuit. 

For  r0  =  .8  and  x  =  0,  x  =  +  .8,  and  x  =  -  .8,  the  polar 
diagrams  are  shown  in  Figs.  37,  38,  39. 


§45] 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


61 


2.)    Reactance  in  series  with  a  circuit. 
45.    In  a  constant  potential  system  of  impressed  E.M.F., 


let  a  reactance,  x0 ,  be  connected  in  series  in  a  receiver  cir- 
cuit of  impedance 

2  =  Vr2  +  *2. 


=  r  —x, 


[100 

90 

• 

IMPE 
^ 

IM 
DANC 
LIN 

PRESSED  E.M.F. 
3E  OF  RECEIVER 

B  RESISTANCE  C 

CONSTANT, 

CIRCUIT  cor 

ONSTA.NT   r 

r, 

EO  " 
4ST 

—    . 

IOO 
XNT, 

2 

8 

z- 

1.0 

X 

VOLTS  AND  AMPER 

g  S  g  §  3  § 

^ 

^ 

-—  ^ 

*B 

- 

•^i      -. 

J± 

—     - 

-    — 

_      

;=— 

—  -= 

^ 

It 

DUC 

TAf 

CE 

RE/ 

CT^ 

NCE 

CON 

DEN 

SAN 

CE- 

a:= 

OHI 

/IS 

o 
Fig.   37. 

+1. 

.0 

.8 

.7 

.6 

.5 

.4 

.8 

.2 

.1 

T-.l    ' 

-.2  - 

-.3  - 

-.4  - 

-.5  - 

-.6   • 

-.7  • 

-.8  - 

-.9-1 

Variation  of    Voltage   at   Constant   Series  Resistance   with   Phase  Relation   of 
Receiver  Circuit. 

Then,  the  total  impedance  of  the  circuit  is 
Z       x  =  r—x      x. 


r   Er 


Fig.  38. 

and  the  current  is, 
f= 


Fig.  39. 


while  the  difference  of  potential  at  the  receiver  terminals 
is, 


62  ALTERNATING-CURRENT  PHENOMENA.  [§45 

Or,  in  absolute  quantities  :  — 
Current, 


E.M.F.  at  receiver  terminals, 


=  £  J      r'  +  x'2       = 

f  V  ra  +  (x  +  xy      v**       2 


2  **. 
difference  of  phase  in  receiver  circuit, 


tan  £  =  -  ; 

x 


difference  of  phase  in  supply  circuit, 


a.)  If  x  is  small  compared  with  r,  that  is,  if  the  receiver 
circuit  is  non-inductive,  /  and  E  change  very  little  for  small 
values  of  x0  ;  but  if  x  is  large,  that  is,  if  the  receiver  circuit 
is  of  large  reactance,  /  and  E  change  much  with  a  change 
of  x0. 

b.)  If  x  is  negative,  that  is,  if  the  receiver  circuit  con- 
tains condensers,  synchronous  motors,  or  other  apparatus 
which  produce  leading  currents  —  above  a  certain  value  of 
x0  the  denominator  in  the  expression  of  E,  becomes  <  z,  or 
E  >  E0  ;  that  is,  the  reactance,  x0  ,  raises  the  potential. 

c.)  E  =  E0  ,  or  the  insertion  of  a  series  inductance,  x0  , 
does  not  affect  the  potential  difference  at  the  receiver  ter- 
minals, if  _ 


or,  x0  =  —  2  x. 

That  is,  if  the  reactance  which  is  connected  in  series  in 
the  circuit  is  of  opposite  sign,  but  twice  as  large  as  the 
reactance  of  the  receiver  circuit,  the  voltage  is  not  affected, 
but  E  =  E0,  /=  E0j  z.  If  x0  >  —  2  x,  it  raises,  if  x  <  —  2x, 
it  lowers,  the  voltage. 

We  see,  then,  that  a  reactance  inserted  in  series  in 
an  alternating-current  circuit  will  lower  the  voltage  at  the 


§45] 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


63 


receiver  terminals  only  when  of  the  same  sign  as  the  reac- 
tance of  the  receiver  circuit  ;  when  of  opposite  sign,  it  will 
lower  the  voltage  if  larger,  raise  the  voltage  if  less,  than. 
twice  the  numerical  value  of^he  reactance  of  the  receiver 
circuit. 

d.)    If  x  =  0,   that    is,    if    the   receiver    circuit    is    non- 
inductive,  the  E.M.F.  at  receiver  terminals  is  : 

£= 


-*{  <-!(?)'+!(-)'-  +  ... 

(1  -\-  x)~*  expanded  by  the  binomial  theorem  is 


1  +  nx  + 


«(»-!) 
1-2 


is  small  compared  with  r :  — 


\Vi  +  * 

(1+*) 

Therefore,  if 


•g.-^g     =    !/*••- 

£0  *\r 

That  is,  the  percentage  drop  of  potential  by  the  insertion 
of  reactance  in  series  in  a  non-inductive  circuit  is,  for  small 


Er,     Ero 


Fig.  40. 


values  of  reactance,  independent  of  the  sign,  but  propor- 
tional to  the  square  of  the  reactance,  or  the  same  whether 
it  be  inductance  or  condensance  reactance. 


ALTERNA  TING-CURRENT  PHENOMENA. 


[§46 


46.  As  an  instance,  in  Fig.  41  the  changes  of  current, 
/,  and  of  E.M.F.  at  receiver  terminals,  E,  at  constant  im- 
pressed E.M.F.,  E0,  are  shown  for  various  conditions  of  a 
receiver  circuit  and  amounts  of  reactance  inserted  in  series. 

Fig.  41  gives  for  various  values  of  reactance,  x0  (if  posi- 
tive, inductance  —  if  negative,  condensance),  the  E.M.Fs., 
£,  at  receiver  terminals,  for  constant  impressed  E.M.F.,. 

VOLTS  E  OR  AMPERES  I 


100 
90 

8. 

J70 

|: 

>30 
20 
10 

Xo   +3 

IMPRESSED  E.'M.F.  CONSTANT,  E 
-IMPEDANCE  OF  RECEIVER  CIRC.UI 

II.   r=.6     £=+.8 
III.   r=.6     X  =  -.8 

5=160 

T  CONS' 

a 

PAN 

>l 

r.z 

=  1. 

ni" 

0 

16 

0 

£ 

^ 

/ 

\ 

u 

o 

/ 

* 

\ 

/ 

\ 

14 

0 

/ 

\ 

/ 

\ 

13 

0 

/ 

\ 

/ 

\ 

12 

0 

/ 

\ 

/ 

\H 

o/ 

/ 

\ 

7 

/ 

^ 

^ 

X 

^ 

X 

/ 

III 

/ 

^ 

/ 

8 

0 

\ 

S 

\ 

/ 

/ 

/ 

y 

? 

7 

0 

> 

V^ 

x, 

/ 

/ 

\s 

/ 

/ 

/ 

| 

0 

> 

X 

X 

X 

^ 

/ 

. 

S 

i 

S 

5 

0 

X 

^ 

.  — 

^ 

^ 

*** 

4 

0 

—  . 

—.    -- 
—     -" 

_-^- 

== 

- 
^-— 

^~~ 



• 

_3 

2 

?_ 

0 

1 

0 

I 

c; 

HM 

S 

IN 

DUOTAr 

CE 

-RE 

AC, 

ANC 

E- 

-^c 

ONI 

)EN 

3ANCE 

0   2.8   2.6   2.4   2  2   2.0    1.8   1.6    1.4    1.2    1.0    .8     .€ 

Fig.  41. 

.4  -f  .2      0  —.2     .4      .6     .8     1.0    1.2-1. 

EQ  =  100  volts,  and   the  following    conditions    of    receiver 

;   «=1.0,  r=1.0,  *=        0  (Curve  I.) 
2=1.0,  r=    .6,*=       .8  (Curve  II.) 
0=1.0,  r=    .6,  x=  -.8  (Curve  III.) 

As  seen,  curve  /  is  symmetrical,  and  with  increasing  x0 
the  voltage  E  remains  first  almost  constant,  and  then  drops 
off  with  increasing  rapidity. 

In  the  inductive  circuit  series  inductance,  or,  in  a  con- 
denser circuit  series  condensance,  causes  the  voltage  to  drop 
off  very  much  faster  than  in  a  non-inductive  circuit. 


§46] 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


65 


Series  inductance  in  a  condenser  circuit,  and  series  con- 
densance  in  an  inductive  circuit,  cause  a  rise  of  potential. 
This  rise  is  a  maximum  for  x0  =  i  .8,  or,  x0  —  —  x  (the 
condition  of  resonance),  and  the  E.M.F.  reaches  the  value, 
E  =  167  volts,  or,  E  =  E0^fr.  This  rise  of  potential  by 
series  reactance  continues  up  to  x0  =  =t  1.6,  or,  XQ  —  —  2x, 


Fig.  42. 

• 

where  E  =  100  volts  again ;  and  for  x0  >  1.6  the  voltage 
drops  again. 

At  x0  =  i  .8,  x  =  =p  .8,  the  total  impedance  of  the  circuit 
is  r  —  j  (x  +  x^  =  r  =  .6,  x  +  x0  =  0,  and  tan  w0  =  0  ; 
that  is,  the  current  and  E.M.F.  in  the  supply  circuit  are 
in  phase  with  each  other,  or  the  circuit  is  in  electrical 
resonance. 


Er 


Fig.  43. 

Since  a  synchronous  motor  in  the  condition  of  efficient 
working  acts  as  a  condensance,  we  get  the  remarkable  result 
that,  in  synchronous  motor  circuits,  choking  coils,  or  reactive 
coils,  can  be  used  for  raising  the  voltage. 

In  Figs.  42  to  44,  the  polar  diagrams  are  shown  for  the 
conditions  — 

EQ  =  100,  x0  =  .6,  x  =        0  (Fig/ 42)  E  =    85.7 

x  =  +  .4  (Fig.  43)  E  =    73.7 

x  ==  —  .4  (Fig.  44)  E  =  106.6 


ALTERNA  TING-CURRENT  PHENOMENA. 


[§47 


47.  In  Fig.  45  the  dependence  of  the  potential,  E,  upon 
the  difference  of  phase,  <o,  in  the  receiver  circuit  is  shown 
for  the  constant  impressed  E.M.F.,  E0  =  100  ;  for  the  con- 
stant receiver  impedance,  z  =  1.0  (but  of  various  phase 
differences  w),  and  for  various  series  reactances,  as  follows  : 


*0=  .2 
x0=  .6 
x,=  -8 
x0  =  1.0 
x9  =  1.6 
x0  =  3.2 


(Curve  I.) 
(Curve  II.) 
(Curve  III.) 
(Curve  IV.) 
(Curve  V.) 
(Curve  VI.) 


Fig.  44. 

Since  z  =  1.0,  the  current,  7,  in  all  these  diagrams  has 
the  same  value  as  E. 

In  Figs.  46  and  47,  the  same  curves  are  plotted  as  in 
Fig.  45,  but  in  Fig.  46  with  the  reactance,  x,  of  the  receiver 
circuit  as  abscissae ;  and  in  Fig.  47  with  the  resistance,  r,  of 
the  receiver  circuit  as  abscissae. 

As  shown,  the  receiver  voltage,  E,  is  always  lowest  when 
x0  and  x  are  of  the  same  sign,  and  highest  when  they  are 
of  opposite  sign. . 

The  rise  of  voltage  due  to  the  balance  of  x0  and  x  is  a 
maximum  for  xa  =  +  1.0,  x  =  —  1.0,  and  r  =  0,  where 


§47] 


RESISTANCE,   INDUCTANCE,    CAPACITY. 


67 


^/|  w=t-90  SO   70  bO  50  40  30  20  10   0   10  20  30  40  50  00  70  bO  90  DEGREES 

Fig.  45.     Variation  of  Voltage  at  Constant  Series  Reactance  with  Phase  Angle  of 
Receiuer  Circuit. 


Fig.  46.     Variation  of  Voltage  at  Constant  Series  Reactance  with  Reactance  of 
Receiuer  Circuit. 


68 


AL  TERNA  TING-CURRENT  PHENOMENA. 


[§48 


E  =  oo  ;  that  is,  absolute  resonance  takes  place.  Obvi- 
ously, this  condition  cannot  be  completely  reached  in 
practice. 

It  is  interesting  to  note,  from  Fig.  47,  that  the  largest 
part  of  the  drop  of  potential  due  to  inductance,  and  rise  to 
condensance  —  or  conversely  —  takes  place  between  r  =  1.0 
and  r  =  .9  ;  or,  in  other  words,  a  circuit  having  a  power 


Fig.  47.     Variation  of  Voltage  at  Constant  Series  Reactance  with  Resistance  of 
Receiver  Circuit. 

factor  of  cos  w  =  .9,  gives  a  drop  several  times  larger  than 
a  non-inductive  circuit,  and  hence  must  be  considered  as  a 
highly  inductive  circuit. 

3.)  Impedance  in  scries  with  a  circuit, 
48.  By  the  use  of  reactance  for  controlling  electric 
circuits,  a  certain  amount  of  resistance  is  also  introduced, 
due  to  the  ohmic  resistance  of  the  conductor  and  the  hys- 
teretic  loss,  which,  as  will  be  seen  hereafter,  can  be  repre- 
sented as  an  effective  resistance. 


§49]  RESISTANCE,  INDUCTANCE,  CAPACITY.  69 

Hence  the  impedance  of  a  reactive  coil  (choking  coil) 
may  be  written  thus  :  — 

.  ^o  =  r0  —jx^  z0  =  Vr02  -f  x02, 

\where  r0  is  in  general  small  Compared  with  XQ  . 
From  this,  if  the  impressed  E.M.F.  is 


.and  the  impedance  of  the  consumer  circuit  is 
Z  =  r  —  jx  =  z  =  V>2  +  x* 

we  get  the  current,     /= —  = * 

Z  +  Z0       (r  +  r.) -/( 
and  the  E.M.F.  at  receiver  terminals, 

E  =  E0  — - —  =  E0 r.—J* 

°  ° 


Or,  in  absolute  quantities, 
the  current  is, 

/=  E° 


the  E.M.F.  at  receiver  terminals  is, 

E  =  £±*—  -  E«z 


+  roy  +  (X  +  ^0)2         V*2  +  *02  +  2  (rr0 
the  difference  of  phase  in  receiver  circuit  is, 


tan  to  =^=  -  ; 
x 


.and  the  difference  of  phase  in  the  supply  circuit  is, 


0  . 

x  +x0 

49.  In  this  case,  the  maximum  drop  of  potential  will  not 
take  place  for  either  x  =  0,  as  for  resistance  in  series,  or 
for  r  =  0,  as  for  reactance  in  series,  but  at  an  intermediate 
point.  The  drop  of  voltage  is  a  maximum  ;  that  is,  E  is 
a  minimum  if  the  denominator  of  E  is  a  maximum  ;  or, 
since  z,  zot  r0,  x0  are  constant  if  rr0  +  xx0  is  a  maximum, 
since  x  =  V^2  —  r2  if  rr0  +  x0  V^2  —  r^  is  a  maximum, 


70 


A  L  TERN  A  TING    CURRENT-PHENOMENA. 


[§  49 


a  function,  f  =  rr0  -f  x0  V^r2  —  r2  is  a  maximum  when 
its  differential  coefficient  equals  zero.  For,  plotting  /  as 
curve  with  r  as  abscissae,  at  the  point  where  /  is  a  maxi- 
mum or  a  minimum,  this  curve  is  for  a  short  distance 
horizontal,  hence  "the  tangens-function  of  its  tangent  equals 
zero.  The  tangens-function  of  the  tangent  of  a  curve,  how- 
ever, is  the  ratio  of  the  change  of  ordinates  to  the  change: 
of  abscissae,  or  is  the  differential  coefficient  of  the  func- 
tion represented  by  the  curve. 


150 
110 

/ 

J30 

/ 

/ 

110 

t 

/ 

/ 

S 

/ 

80 

ZP 

60 

j^--- 

^-* 

^ 

^  — 

i— 

•      — 

—      — 

Zo  = 

^-"•••" 

3^ 

—  — 

-  —  ' 

„  

^^•^ 

J 

60 

^ 

*• 

-""^ 

—  ~  •* 

-    ~* 

.  

• 

,  - 

^^ 

30 
20 

—  H-, 

—     <• 

i 

_ZA 

-i.fc 

-  1  . 

10 
0 

X- 

—  >• 

1. 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

- 

-.1    - 

-.2  - 

-.3   - 

-.4  - 

-.§  - 

-.6   • 

-.7  • 

-.8  • 

T9-1. 

Fig.  48. 

Thus  we  have  :  — 

f  =  rr0  -f-  x0  -\/z  2  —  r2  =  maximum  or  minimum,  if 


Differentiating,  we  get :  — 


§50] 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


71 


That  is,  the  drop  of  potential  is  a  maximum,  if  the  re- 
actance factor,  x I r,  of  the  receiver  circuit  equals  the  reac- 
tance factor,  x0l 'r0)  of  the  series  impedance. 


Fig.  49. 


I, 
lo 

Fig.  50. 


50.  As  att  example,  Fig.  48  shows  the  E.M.F.,  E, 
at  the  receiver  terminals,  at  a  constant  impressed  E.M.F., 
E0  =  100,  a  constant  impedance  of  the  receiver  circuit,. 
z  =  1.0,  and  constant  series  impedances, 

Z  =    .3  -y.4  (Curve  I.) 

Z0  =  1.2  —  yl.6         (Curve  II.) 
as  functions  of  the  reactance,  ;r,  of  the  receiver  circuit 


Fig.  57. 


Figs.    49  to  51  give  the  polar  diagram  for  E0  =  100, 
x  =  .9,  x  =  0,  ^r  =  -  .9,  and  Z0  =  .3  -7  .4. 


ALTERNATING-CURRENT  PHENOMENA. 


51 


4.)    Compensation   for   Lagging    Currents    by    Shunted 
Condensance. 

51.  We  have  seen  in  the  latter  paragraphs,  that  in  a 
constant  potential  alternating-current  system,  the  voltage 
at  the  terminals  of  a  receiver  circuit  can  be  varied  by  the 
use  of  a  variable  reactance  in  series  to  the  circuit,  without 
loss  of  energy  except  the  unavoidable  loss  due  to  the 
resistance  and  hysteresis  of  the  reactance ;  and  that,  if 
the  series  reactance  is  very  large  compared  with  the  resis- 
tance of  the  receiver  circuit,  the  current  in  the  receiver 
circuit  becomes  more  or  less  independent  of  the  resis- 
tance,—  that  is,  of  the  power  consumed  in  the  receiver 


Fig.  52. 


-circuit,  which  in  this  case  approaches  the  conditions  of  a 
constant  alternating-current  circuit,  whose  current  is. 


I 


,   or  approximately,  /  =  — ° . 


This  potential  control,  however,  causes  the  current  taken 
from  the  mains  to  lag  greatly  behind  the  E.M.F.,  and 
thereby  requires  a  much  larger  current  than  corresponds 
to  the  power  consumed  in  the  receiver  circuit. 

Since  a  condenser  draws  from  the  mains  a  leading  cur- 
rent, a  condenser  shunted  across  such  a  circuit  with  lagging 
current  will  compensate  for  the  lag,  the  leading  and  the 
lagging  current  combining  to  form  a  resultant  current  more 
or  less  in  phase  with  the  E.M.F.,  and  therefore  propor- 
tional to  the  power  expended. 


§52]  RESISTANCE,  INDUCTANCE,  CAPACITY.  73' 

In  a  circuit  shown  diagrammatically  in  Fig.  52,  let  the 
non-inductive  receiver  circuit  of  resistance,  r,  be  connected 
in  series  with  the  inductance,^,  and  the  whole  shunted  by 
a  condenser  of  condensance*  c,  entailing  but  a  negligible  loss. 
of  energy. 

Then,  if  E0  =  impressed  E.M.F.,  - 

the  current  in  receiver  circuit  is, 


the  current  in  condenser  circuit  is, 


and  the  total  current  is 


r  —  jx0      jc 


2 

or,  in  absolute  terms,  I0  =  E0\I (  —^ -}  +  (        •** 1  ' 


while  the  E.M.F.  at  receiver  terminals  is, 

E0r 


^  = 


52.  The  main  current,  70,  is  in. phase  with  the  impressed 
E.M.F.,  E0,  or  the  lagging  current  is  completely  balanced^ 
or  supplied  by,  the  condensance,  if  the  imaginary  term  in 
the  expression  of  I0  disappears  ;  that  is,  if 


r2  + 
This  gives,  expanded  : 

*o 

Hence  the  capacity  required  to  compensate  for  the 
lagging  current  produced  by  the  insertion  of  inductance 
in  series  to  a  non-inductive  circuit  depends  upon  the  resis- 
tance and  the  inductance  of  the  circuit.  x0  being  constant,. 


74  ALTERNATING-CURRENT  PHENOMENA.  [§52 

with  increasing  resistance,  r,  the    condensance   has   to   be 
increased,  or  the  capacity  decreased,  to  keep  the  balance. 

~2  _|_T2 

Substituting  c  =  -        ^-  , 

Xo 

we  get,  as  the  equations  of  the  inductive  circuit  balanced 
by  condensance  :  — 


r  —  jx0  r2-  -+-  xj  Vr2  +  x, 

j  =  _     jE0x0  j  =     E0xc 

I    =  £°r  I    =  £or 


and  for  the  power  expended  in  the  receiver  circuit :  — 
72  ^'  _   £<?r     -  T  p 

-^+^-'°£» 

that  is,  the  main  current  is  proportional  to  the  expenditure 
of  power. 

For  r  =  0  we  have  x  =  x0,  or  the  condition  of  balance. 

Complete  balance  of  the  lagging  component  of  current 
by  shunted  capacity  thus  requires  that  the  condensance,  c, 
be  varied  with  the  resistance,  r;  that  is,  with  the  varying 
load  on  the  receiver  circuit. 

In  Fig.  53  are  shown,  for  a  constant  impressed  E.M.F., 
E0  =  1000  volts,  and  a  constant  series  reactance,  x0  =  100 
ohms,  values  for  the  balanced  circuit  of, 

current  in  receiver  circuit       (Curve  I.), 
current  in  condenser  circuit  (Curve  II.), 
current  in  main  circuit  (Curve  III.), 

E.M.F.  at  receiver  terminals  (Curve  IV.), 

with  the  resistance,  r,  of  the  receiver  circuit  as  abscissae. 


§53] 


RESISTANCE,   INDUCTANCE,    CAPACITY. 


75 


IMPRESSED  E.M.F.  CONSTANT,  E0  =  IOOO  VOLTS. 
SERIES    REACTANCE  -CONSTANT,  X0=  IOO  OHMS. 
VARIABLE  RESISTANCE  IN  RECEIVER  CIRCUIT. 
BALA.NCED  BY  VARYING  THE  SHUNTED  CONDENSANCE, 

I.    CURRENT  IN  RECEIVER  CIRCUIT. 

II.  CURRENT  IN  CONDENSER  CIRCUIT. 

III.  CURRENT  IN  MAIN  CIRCUIT. 
IV."  E.M.F.  AT  REQEIVER  CIRCUIT. 


RESISTANCE- *-r,  OF  RECEIVER  CIRCUIT    OHMS 
10    20    30    40     50    60     70     80    90    100   110  120  130  140    150  160  170  180   190  200 


Fig,  53.    Compensation  of  Lagging  Currents  in  Receiving  Circuit  by  Variable  Shunted 

Condensance. 

53.  If,  however,  the  condensance  is  left  unchanged, 
c  =  x0  at  the  no-load  value,  so  that  if  the  circuit  is  balanced 
for  r  =  0,  it  will  be  overbalanced  for  r  >  0,  and  the  main 
current  will  become  leading. 

We  get  in  this  case  :  — 


The  difference  of  phase  in  the  main  circuit  is,  — 


tan  a>rt  =  — 


0 


76 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


[§54 


when  r  —  0  or  at  no  load, '  and  increases  with  increasing 
resistance,  as  the  lead  of  the  current.  At  the  same  time,,, 
the  current  in  the  receiver  circuit,  /,  is  approximately  con- 
stant for  small  values  of  r,  and  then  gradually  decreases. 


/• 

AMPERES 
10 

fl 

8 
7 
6 
6 
4 
3 
2 
^l 

IMPRESSED  E.M.F.  CONSTANT,  E0=IOOO  VOLTS. 
SERIES   REACTANCE    CONSTANT,  X0  —  JOO  OHMS. 
SHUNTED  CONDENSANCE  CONSTANT,    C=  IOO  OH 
VARIABLE  RESISTANCE.  IN  RECEIVER  CIRCUIT- 

(.CURRENT  IN  RECEIVER  CIRCUIT. 
II.  CURRENT  IN  CONDENSER  CIRCUIT. 
III.  CURRENT  IN  MAIN  CIRCUIT. 
IV.E.M.F.  AT  RECEIVER  CIRCUIT. 

MS. 

VOL 

6 

II. 

1<XK 
900 

—  —  . 

'  „ 

=^ 

^ 

^^ 

.^*—  - 



.  —  1 

,•—  - 

—      — 

iS—  - 

= 

i—  — 

> 

V 

f^~ 

-^- 

-600 
-500 

± 

^ 

**"^' 

^^**^ 

•--, 

~-^ 

IV^ 

/ 

—  .. 

"*-  

•—^ 

III, 

£ 

/ 

--—  *. 

-*-*. 

*—  "^ 

-•  —  . 

/ 

/ 

r-100 

/ 

RESISTANCE  r—  OF  RECEIVER  CIRCUIT,  OHMS. 

7 

1 

1  1  1 

10  20  30  40  50  60  .70  80  90  100  110  120  130  140  150  160  170  180  190  200  OH  MS 
Fig.  54. 

In  Fig.  54  are  shown  the  values  of  /,  /!,/<>,  E,  in  Curves 
I.,  II.,  III.,  IV.,  similarly  as  in  Fig.  50,  for  E0  =  1000  volts, 
c  =  XQ  =  100  ohms,  and  r  as  abscissae. 


5.)    Constant  Potential —  Constant  Current   Transformation. 

54.  In  a  constant  potential  circuit  containing  a  large 
and  constant  reactance,  xlt  and  a  varying  resistance,  r,  the 
current  is  approximately  constant,  and  only  gradually  drops 
off  with  increasing  resistance,  r,  —  that  is,  with  increasing 
load,  —  but  the  current  lags  greatly  behind  the  E.M.F.  This 
lagging  current  in  the  receiver  circuit  can  be  supplied  by  a 
shunted  condensance.  Leaving,  however,  the  condensance 
constant,  c  =  x0,  so  as  to  balance  the  lagging  current  at  no 


§54] 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


77 


load,  that  is,  at  r  =  0,  it  will  overbalance  with  increasing 
load,  that  is,  with  increasing  r,  and  thus  the  main  current 
will  become  leading,  while  the  receiver  current  decreases 
if  the  impressed  E.M.F.,  E*,  is  kept  constant.  Hence,  to 
keep  the  current  in  the  receiver  circuit  entirely  constant,  the 
impressed  E.M.F.,  E0,  has  to  be  increased  with  increasing 
resistance,  r;  that  is,  with  increasing  lead  of  the  main  cur- 
rent. Since,  as  explained  before,  in  a  circuit  with  leading 
current,  a  series  inductance  raises  the  potential,  to  maintain 
the  current  in  the  receiver  circuit  constant  under  all  loads, 
an  inductance,  x^ ,  inserted  in  the  main  circuit,  as  shown  in 
the  diagram,  Fig.  55,  can  be  used  for  raising  the  potential, 
E0,  with  increasing  load. 


Fig.  55. 


Let  — 


be  the  impressed  E.M.F.  of  the  generator,  or  of  the  mains, 
and  let  the  condensance  be  xc  =  x0  ;  then  — 
Current  in  receiver  circuit, 


r  —x0 


current  in  condenser  circuit, 


Hence,  the  total  current  in  main  line  is 


Er 


r—jx0 


78  ALTERNATING-CURRENT  PHENOMENA.      £   [§  54 

and  the  E.M.F.  at  receiver  terminals, 

E  =  Ir  =     E'r   ; 

r— /*<, 

E.M.F.  at  condenser  terminals, 

^o5 
E.M.F.  consumed  in  main  line, 


hence,  the  E.M.F.  at  generator  is 

E*=E0  +  E'=En  )l AX 


and  the  E.M.F.  at  condenser  terminals, 


current  in  receiver  circuit, 


r  —jx0       r(x0  —  xz)—  jx? 


This  value  of  /  contains  the  resistance,  r,  only  as  a  fac- 
tor to  the  difference,  x0  —  ;r2  ;  hence,  if  the  reactance,  ;r2  , 
is  chosen  =  x0  ,  r  cancels  altogether,  and  we  find  that  if 
x^  =  x0  ,  the  current  in  the  receiver  circuit  is  constant, 

/=y-2, 

*0 

and  is  independent  of  the  resistance,  r  ;   that  is,  of  the  load. 

Thus,  by  substituting,  we  have,  ;r2  =  x0,  - 
Impressed  E.M.F.  at  generator, 


£2  =  ^2  +  /V,  ^2  =  V^22  +  *2' 2  =  constant ; 

current  in  receiver  circuit, 

7    =y^-,  /  =  ^  =  constant ; 

x0  x0 

E.M.F.  at  receiver  circuit, 

E  =Sr=/^^-,        E  =  E*L    or  proportional  to  load  r; 


55]  RESISTANCE,  INDUCTANCE,  CAPACITY. 


79 


E.M.F.  at  condenser  terminals, 
E0  =  £*  (x°  +Jr^ 

*0 

.    w1+yrw.L 

\  -w 

current  in  condenser  circuit, 


-l,  hence  >  Ez ; 


main  current, 

Enr 


£zr  E  r  (  ProP°rtional to the  load, 

=  ~TT »  S0  =  -—  ,  <.  r,    and    in    phase   with 

0  (E.M.F,  E,. 

The  power  of  the  receiver  circuit  is, 


IE 


the  power  of  the  main  circuit, 


,  hence  the  same. 


55.    This  arrangement  is  entirely  reversible  ;  that  is, 
if  EZ  =  constant,  7    =  constant  ;  and 
if  -4    =  constant,  E  =  constant. 

In  the  latter  case  we  have,  by  expressing  all  the  quanti- 
ties by  f0  :  - 
Current  in  main  line, 

fQ   =  constant; 
E.M.F.  at  receiver  circuit, 

£   =  So*,,  —  constant  ; 
current  in  receiver  circuit, 


—  ,  proportional  to  the  load  i; 


current  in  condenser  circuit, 


80  ALTERNATING-CURRENT  PHENOMENA.  [§55 

E.M.F.  at  condenser  terminals, 


Impressed  ElM.F.  at  generator  terminals, 

x 2  1 

Ez  =  — /0 ,  or  proportional  to  the  load  -  . 

From  the  above  we  have  the  following  deduction  : 

Connecting  two  reactances  of  equal  value,  x0,  in  series 
to  a  non-inductive  receiver  circuit  of  variable  resistance,  ry 
and  shunting  across  the  circuit  from  midway  between  the 
inductances  by  a  capacity  of  condensance,  xc  =  x0,  trans- 
forms a  constant  potential  main  circuit  into  a  constant  cur- 
rent receiver  circuit,  and,  inversely,  transforms  a  constant 
current  main  circuit  into  a  constant  potential  receiver  cir- 
cuit. This  combination  of  inductance  and  capacity  acts  as 
a  transformer,  and  converts  from  constant  potential  to  con- 
stant current  and  inversely,  without  introducing  a  displace- 
ment of  phase  between  current  and  E.M.F. 

It  is  interesting  to  note  here  that  a  short  circuit  in  the 
receiver  circuit  acts  like  a  break  in  the  supply  circuit,  and  a 
break  in  the  receiver  circuit  acts  like  a  short  circuit  in  the 
supply  circuit. 

As  an  instance,  in  Fig.  56  are  plotted  the  numerical 
values  of  a  transformation  from  constant  potential  of  1,000 
volts  to  constant  current  of  10  amperes. 

Since  Ez  =  1,000,  /  =  10,  we  have  :  x0  =  100  ;  hence 
the  constants  of  the  circuit  are  :  — 

E*  =  1000  volts  ; 

/    =  10  amperes; 

E   =  10  r,  plotted  as  Curve  I.,  with  the  resistances,/-,  as  abscissae; 

/         /   *-  \2 

E0  =  1000  V/l  +  (  -y-  ),  plotted  as  Curve  II. ; 
V  100  I 

/!  =  10 y/1  +  (— Y,  plotted  as  Curve  III.- 
70   =  .1  r,  plotted  as  Curve  IV. 


§56] 


RESISTANCE,  INDUCTANCE,  CAPACITY. 


81 


56.  In  practice,  the  power  consumed  in  the  main  circuit 
will  be  larger  than  the  power  delivered  to  the  receiver  cir- 
cuit, due  to  the  unavoidable  losses  of  power  in  the  induc- 
tances and  condensances. 


CURRENT  IN  RECEIVER  CIRCUIT  CONSTANT, 
IMPRESSED  E.M.F.CONSTANT,  E2=IOOO  VOL 
2    REACTANCES   OF  X0  =IOO  OHMS  EACH.  SH 
THE  CONDENSANCE,  ZC  =  IOO  OHMS. 
VARIABLE  RESISTANCE  IN  RECEIVER  CIRCUI 
1.    E.M.F.  AT  RECEIVER  CIRCUIT. 
5         II,    E.M-F.  AT  CONDENSER  CIRCUIT. 
III.  CURRENT  IN  CONDENSER  CIRCUIT. 
IV.  CURRENT  IN  MAIN  LINE 
V.  CURRENT  IN  MAIN  LINE  INCLUDING  LC 
VI.   EFFICIENCY  OF  TRANSFORMATION. 

1  -=>IO  AMPERES 
T.S, 
UNTED  IN  THE.IR  IV 

r. 

90 
80 
70 
60  o 

— 

IDC 

LE 

BY 

OLT 

1400 

x-** 

^ 

1300 

5SSE 

S                  ^ 

^-*- 

—  ' 

" 

/ 

12CO 

J^^^ 

--^ 

w^-> 

s 

' 

1100 

_\\i 

^ 

-—• 

*~*'^ 

£ 

1000 
900 

—     — 

i    — 

r=^ 

.—  • 

»«—  • 

-—  — 

VI 

T: 

r^ 

- 

x^ 

^L 

X* 

800 

^ 

•-^ 

"^ 

y 

•'' 

^ 

^ 

700 

/ 

Nl 

X'' 

^ 

^ 

600/ 

f  ^ 

' 

\> 

X' 

50/0 

, 

''' 

x^ 

x^ 

r 

.''' 

^< 

^ 

in  M 

3/00 

x 

X 

^ 

30 
20 
10 

200 

^-' 

'^X 

^ 

100 

S 

^ 

^ 

=iES 

ST/S 

NCE 



r  c 

IF   R 

ECE 

IVE 

=?  Cl 

RCL 

IT, 

OHf 

us 

^X 

^ 

Fig.  56.    Constant-Potential  —  Constant-Current  Transformation. 

Let- 

r\  =  2  ohms  =  effective  resistance  of  condensance  ; 

r0  =  3  ohms  =  effective  resistance  of  each  of  the  inductances. 

We  then  have  :  — 

Power  consumed  in  condensance,  I?  r±  =  200  +  .02  r2  ; 
power  consumed  by  first  inductance,  72  r0  =  300  ; 
power  consumed  by  second  inductance,  702  r0  =  .03  r2. 
Hence,  the  total  loss  of  energy  is  500  -f-  -05  r2  ; 
output  of  system,  /2  r  =  100  r 

input,  500  +  100r  +  .05r2; 

100  r 


It  follows  that  the  main  current,  I0,  increases  slightly 
by  the  amount  necessary  to  supply  the  losses  of  energy 
in  the  apparatus. 


82  ALTERNATING-CURRENT  PHENOMENA.  [§56 

This  curve  of  current,  70,  including  losses  in  transforma- 
tion, is  shown  in  dotted  lines  as  Curve  V.  in  Fig.  56  ;  and 
the  efficiency  is  shown  in  broken  line,  as  Curve  VI.  As 
shown,  the  efficiency  is  practically  constant  within  a  wide 
range. 


§57]  RESISTANCE    OF   TRANSMISSION  LINES.  83 


CHAPTER    IX. 

RESISTANCE   AND   REACTANCE    OF   TRANSMISSION   LINES. 

57.  In  alternating-current  circuits,  E.M.F.  is  consumed 
in  the  feeders  of  distributing  networks,  and  in  the  lines  of 
long-distance  transmissions,  not  only  by  the  resistance,  but 
also  by  the  reactance,  of  the  line.  The  E.M.F.  consumed  by 
the  resistance  is  in  phase,  while  the  E.M.F.  consumed  by  the 
reactance  is  in  quadrature,  with  the  current.  Hence  their 
influence  upon  the  E.M.F.  at  the  receiver  circuit  depends 
upon  the  difference  of  phase  between  the  current  and  the 
E.M.F.  in  that  circuit.  As  discussed  before,  the  drop  of 
potential  due  to  the  resistance  is  a  maximum 'when  the 
receiver  current  is  in  phase,  a  minimum  when  it  is  in 
quadrature,  with  the  E.M.F.  The  change  of  potential  due 
to  line  reactance  is  small  if  the  current  is  in  phase  with 
the  E.M.F.,  while  a  drop  of  potential  is  produced  with  a 
lagging,  and  a  rise  of  potential  with  a  leading,  current  in 
the  receiver  circuit. 

Thus  the  change  of  potential  due  to  a  line  of  given  re- 
sistance and  inductance  depends  upon  the  phase  difference 
in  the  receiver  circuit,  and  can  be  varied  and  controlled 
by  varying  this  phase  difference ;  that  is,  by  varying  the 
admittance,  Y  =  g  +jb,  of  the  receiver  circuit. 

The  conductance,  g,  of  the  receiver  circuit  depends  upon 
the  consumption  of  power,  —  that  is,  upon  the  load  on  the 
circuit,  —  and  thus  cannot  be  varied  for  the  purpose  of  reg- 
ulation. Its  susceptance,  b,  however,  can  be  changed  by 
shunting  the  circuit  with  a  reactance,  and  will  be  increased 
by  a  shunted  inductance,  and  decreased  by  a  shunted  con- 
densance.  Hence,  for  the  purpose  of  investigation,  the 


84  ALTERNATING-CURRENT  PHENOMENA.  [§58 

receiver  circuit  can  be  assumed  to  consist  of  two  branches, 
a  conductance,  gt  —  the  non-inductive  part  of  the  circuit,  — 
shunted  by  a  susceptance,  b,  which  can  be  varied  without 
expenditure  of  energy.  The  two  components  of  current 
can  thus  be  considered  separately,  the  energy  component  as 
determined  by  the  load  on  the  circuit,  and  the  wattless 
component,  which  can  be  varied  for  the  purpose  of  regu- 
lation. 

Obviously,  in  the  same  way,  the  E.M.F.  at  the  receiver 
circuit  may  be  considered  as  consisting  of  two  components, 
the  energy  component,  in  phase  with  the  current,  and 
the  wattless  component,  in  quadrature  with  the  current. 
This  will  correspond  to  the  case  of  a  reactance  connected 
in  series  to  the  non-inductive  part  of  the  circuit.  Since  the 
effect  of  either  resolution  into  components  is  the  same  so 
far  as  the  line  is  concerned,  we  need  not  make  any  assump- 
tion as  to  whether  the  wattless  part  of  the  receiver  circuit 
is  in  shunt,  or  in  series,  to  the  energy  part. 

Let- 
Z0  =  r0  —  jx0  =  impedance  of  the  line  ;        

o-       ~\/  •}*  ^      I       -y  2  . 

"o  "ft     ~T~  "^o    > 

y  =  g  +  jb    =  admittance  of  receiver  circuit ; 

E0  =  e0  -\-  je0'  =  impressed  E.M.F.  at  generator  end  of  line  ; 

-E0  =  V^02  +  e0 ' 2 ; 
E   —  e    -\-je'   =  E.M.F.  at  receiver  end  of  line; 


E  =  Ve*  +  e'2' 


f0    =  i0  -f  ji0'  =  current  in  the  line  ;  I0  =  V*02  +  i0 ' 2- 

The  simplest  condition  is  that  of  a  non-inductive  receiver 
circuit,  such  as  a  lighting  circuit. 

1.)    Non-inductive  Receiver  Circuit  Stipplied  over  an 

Inductive  Line. 

58.    In  this  case,  the  admittance  of  the  receiver  circuit 
is  Y  =  g,  since  b  =  0. 


$58]  RESISTANCE   OF   TRANSMISSION  LINES.  85 

We  have  then  —  : 

current,  I0  =  Eg', 

impressed  E.M.F.,  £.'=  E  +  ZJ0  =  E  (1  +  Z9g). 

Hence  - 
E.M.F.  at  receiver  circuit, 

E  E°       =  £° 

1  +  Z0g 


•current,  f0  =          *          =  E°g 


Hence,  in  absolute  values  — 
E.M.F.  at  receiver  circuit,  E  = 

current,  I0  = 


The  ratio  of  E.M.Fs.  at  receiver  circuit  and  at  genera- 
tor, or  supply  circuit,  is  — 

E  1 


r'2  „    2 


and  the  power  delivered  in  the  non-inductive  receiver  cir- 
cuit, or 

r?   o   _. 

output,  P  =  I0E  =  — 


As  a  function  of  g,  and  with  a  given  E0,  r0,  and  x0,  this 
power  is  a  maximum,  if  — 

^=0; 

dg- 
that  is  — 

-  1  +  g*r*  +  g*x*  =  0  ; 
hence  — 

conductance  of  receiver  circuit  for  maximum  output, 
=  1  =_  1_ 

"  Vr02  +  X*         *o    ' 

Resistance  of  receiver  circuit,     rm  =  —  =  z0 ; 


86  ALTERNATING-CURRENT  PHENOMENA.  [§59 

and,  substituting  this  in  P  - 

Ma™ 


and  — 

ratio  of  E.M.F.  at  receiver  and  at  generator  end  of  line, 

=  !L  =  1 

a™  ~  E~~ 


efficiency, 


That  is,  the  output  which  can  be  transmitted  over  an 
inductive  line  of  resistance,  r0  ,  and  reactance,  x0  ,  —  that  is, 
of  impedance,  z0  ,  —  into  a  non-inductive  receiver  circuit,  is 
a  maximum,  if  the  resistance  of  the  receiver  circuit  equals 
the  impedance  of  the  line,  r  =  z0,  and  is  — 

77  2 

P   =  ° 

~2(ra  +  ,.)  ' 

The  output  is  transmitted  at  the  efficiency  of 


and  with  a  ratio  of  E.M.Fs.  of 

1 


59.  We  see  from  this,  that  the  maximum  output  which 
can  be  delivered  over  an  inductive  line  is  less  than  the 
output  delivered  over  a  non-inductive  line  of  the  same 
resistance  —  that  is,  which  can  be  delivered  by  continuous 
currents  with  the  same  generator  potential. 

In  Fig.  57  are  shown,  for  the  constants 

E0  =  1000  volts, 

Z0  =  2.5  —  6/ ;  that  is,  r0  =  2.5  ohms,  x0  =  6  ohms,  z0  =  6.5  ohms, 

with  the  current  In  as  abscissae,  the  values  — 


§60] 


RESISTANCE    OF   TRANSMISSION  LINES. 


87 


E.M.F.  at  Receiver  Circuit,  E,  (Curve  I.) ; 

Output  of  Transmission,  P,  (Curve  II.); 

Efficiency  of  Transmission,  (Curve  III.). 

The  same  quantities,  E  arili  P,  for  a  non-inductive  line  of 
resistance,  r0  =  2.5  ohms,  XQ  —  0,  are  shown  in  Curves  IV.,, 
V.,  and  VI. 


NON-INDUCTIVE  RECEIV 
SUPPLIED  OVER  INDUCTIVE  LI 
Z0=2.5-6/ 
AND  OVER  NON-INDUCTIVE  L^ 

r0  =  2.5 

CURVE  1.    E.M.F.  AT  RECEIVER  CIRC 
||      II.'    OUTPUT  IN         » 

"      III.    EFFICIENCY  OF  TRANSMISJ 
•  '     VU 

ER  CIRCUIT 
ME  OF  IM.PEC 

E  OF  RESIS" 

UIT,  INDUCTIV 
NON-INDUCTIV 
INDUCTIV 
NON-INDUCTIV 
ION,  INDUCTIV 
NON-4WDyCTIV 

)AN 

FAN 

E  L^ 
E  ' 
E.  ' 

E  ( 
E  ' 
E  ' 

/ 

:E 

CE 
E 

X 

u 

100 

90 
80 
70 
60 
50 

„ 

UUK 

^* 

^ 



<A 

^ 

z 
-o- 

X 

-H- 

-1- 

0 

5 
w 

• 

/ 

< 

DC 

cc 

LJ 

/ 

~l- 
fc 

UJ 

o 

or 

/ 

>• 

o 

/ 

z 

Ul 

< 
w 

/ 

/ 

U- 

u_ 

-5- 

1000 

/ 

.X'' 

^ 

^ 

<' 

UT 

100^ 

-^ 

^ 

^_^ 

// 

/ 

s 

, 

903: 

900 

40 
30 

20 
10 

^^ 

^ 

, 

~^~~^ 

\-\f 

\ 

30* 

800 

// 

\, 

"^ 

l^. 

-^ 

Jl/ 

\ 

70tf 

TOO 

/ 

> 

^V 

—  . 

£V 

fiOO 

) 

/ 

^ 

\ 

\ 

5(K 

--* 
500 

-~-^ 

/ 

\ 

\ 

m 

400 

/ 

\ 

\ 

3W 

:?oo 

/ 

\ 

\ 

20^ 

:WO 

/ 

\ 

\(H 

100 

/ 

CUF 

RE 

MTI 

N  L 

NE:|  |0 

\MP 

ERE 

3 

\ 

0     10     20     30     40     60     60     70     80     90    100    110  120   130  140  150   160   170  180  ' 
Fig.  57.     Non-inductive  Receiver  Circuit  Supplied  Over  Inductive  Line. 

2.)    Maximum  Power  Supplied  over  an  Inductive  Line. 

60.  If  the  receiver  circuit  contains  the  susceptance,  £r 
in  addition  to  the  conductance,  g,  its  admittance  can  be 
written  thus:  — 


FZ0). 


Then- 

current,  f0  = 

Impressed  E.M.F.,    E0  =  E  +  I0Z0  =  E  (1 


•88  AL  TERN  A  TING-CURRENT  PHENOMENA.  [§61 

Hence  — 

E.M.F.  at  receiver  terminals, 
E  = 


YZ0        (1  +  r0g  +  *0b)-  j  (Xog  _  r.  J)  ' 
•current, 


1  +  YZ0        (1  +  r.g  +  x.b)~  j  (Xog  -r0b}' 
or,  in  absolute  values  — 
E.M.F.  at  receiver  circuit, 


current, 


ratio  of  E.M.Fs.  at  receiver  circuit  and  at  generator  circuit, 


rog  +  ^0  ^)*  +  (x9g  - 
and  the  output  in  the  receiver  circuit  is, 


61.  a.)  Dependence  of  the  output  upon  the  susceptance  of 
the  receiver  circuit. 

At  a  given  conductance,  g,  of  the  receiver  circuit,  its 
•output,  P  =  E^a^g,  is  a  maximum,  if  a2  is  a  maximum  ;  that 
is,  when  — 

/=  ~  =  (1  4-  rog  +  *0bY  +  (*0g  -  r0b? 
is  a  minimum. 

The  condition  necessary  is  — 


or,  expanding, 

Hence  — 
Susceptance  of  receiver  circuit, 

b=  -         *° 


-or  ^  +  b0  =  0, 


§  62]  RESISTANCE   OF   TRANSMISSION  LINES.  89s 

that  is,  if  the  sum  of  the  susceptances  of  line  and  of  receiver 
circuit  equals  zero. 

Substituting  this  value,  we  get  — 

ratio  of  E.M.Fs.  at  maximum  output, 

^  =  ^t  1  . 

£0     z0  (g  +  go} ' 

maximum  output, 

P\  = — ) 

*?  (g  +  goY 

current, 


*0boy  +  (r0b 
and,  expanding, 


phase  difference  in  receiver  circuit, 

~        b  b0 

tan  o>  =  -  =  --  -  ; 

^  ^ 

phase  difference  in  generator  circuit, 


+ 


' 


62.    #.)    Dependence  of  the  output  upon  the  conductance 
of  the  receiver  circuit. 

At  a  given  susceptance,  b,  of  the  receiver   circuit,  its 
output,  P  =  E02a2g,  is  a  maximum,  if  — 


+  r.  g-  +  ^o^)2  +  (*.f  - 


90  ALTERNATING-CURRENT  PHENOMENA.  [§63 

that  is,  expanding,  - 

(1  +  r0g  +  x0l>Y  +  (Xog  -  r0by  -  2g(r0  +  r0*g  +  x0*g)  =  0; 
or,  expanding,  — 

(b  H-  b0Y  =  g*  _  #»;        g  =  V^o2  +  (b  +  boy. 
Substituting  this  value  in  the  equation  for  a,  page  88, 
we  get  - 
ratio  of  E.M.Fs., 


(b 


^o  V2  g  (g  +  ^0)       V2 
power, 


2 


2  {go 


As  a  function  of  the  susceptance,  b,  this  power  becomes 
a  maximum  for  dP^j  db  —  0,  that  is,  according  to  §  61,  if  — 

*•-•—*.-• 

Substituting  this  value,  we  get  — 

^  =  —  ^?<r  =  <ro>7=7oJ  hence:    K=  g  +  jb  =  g0  —  Jb0; 
x  =  —  x0 ,  r  =  r0 ,  ar  =  z0,  Z  =  r  —jx  =  r0  +  jx0  ; 

substituting  this  value,  we  get  — 

ratio  of  E.M.Fs.,  a  m  =  -^  =  £-  ; 

2^      2r0 

power,  ^  =  — -  ; 

"*  ro 

that  is,  the  same  as  with  a  continuous-current  circuit ;  or, 
in  other  words,  the  inductance  of  the  line  and  of  the  receiver 
circuit  can  be  perfectly  balanced  in  its  effect  upon  the 
output. 

63.    As  a  summary,  we  thus  have  : 

The  output    delivered  over  an    inductive  line  of    impe- 


§63]  RESISTANCE   OF   TRANSMISSION  LINES.  91 

dance,  Z0  =  r0  —jx0 ,  into  a  non-inductive  receiver  circuit,  is 
a  maximum  for  the  resistance,  r  =  z0,  or  conductance,  g  = 
yQ ,  of  the  receiver  circuit,  or  — 

•  77  2 

r>  __    t         ^o 

~  2  fro  +  *o)  ' 

at  the  ratio  of  potentials, 


With  a  receiver  circuit  of  constant  susceptance,  b,  the  out- 
put, as  a  function  of  the  conductance,  g,  is  a  maximum  for 
the  conductance,  — 

g  =  Vfra+(*  +  4,)a, 
and  is 

E*y* 

2  (*  +  *,)' 
at  the  ratio  of  potentials, 


With  a  receiver  circuit  of  constant  conductance,  g,  the 
output,  as  a  function  of  the  reactance,  b,  is  a  maximum  for 
the  reactance,  b  —  —  b0,  and  is 

p_  * 

at  the  ratio  of  potentials, 


y0  (g  +  go) 

The  maximum  output  which  can  be  delivered  over  an  in- 
ductive line,  as  a  function  of  the  admittance  or  impedance 
of  the  receiver  circuit,  takes  place  when  Z  —  r0  -f  jx0 ,  or 
Y=  g0—  jb0\  that  is,  when  the  resistance  or  conductance 
of  a  receiver  circuit  and  line  are  equal,  the  reactance  or  sus- 
ceptance of  the  receiver  circuit  and  line  are  equal  but  of 
opposite  sign,  and  is,  P  =  £02  /  4  r0 ,  or  independent  of  the 
reactances,  but  equal  to  the  output  of  a  continuous-current 


92 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


[§64 


circuit  of  equal  line  resistance.  The  ratio  of  potentials  is,  in 
this  case,  a  =  zo  j  2  r0,  while  in  a  continuous-current  circuit 
it  is  equal  to  ^.  The  efficiency  is  equal  to  50  per  cent. 


RATIO  OF  POTENTIAL  nj  ATlREqEIVJNG  ANC 


.01    .02   .03    .Oi    .05    .06   .07     .08    .09  .10    .11    .12    .13    .14    .15    J6    .17 
Fig.  58.     Variation  of  the  Potential  in  Line  at  Different  Loads. 

64.    As   an   instance,   in    Fig.   58    are    shown,    for    the 
constants  — 

E0  =  1000  volts,  and  Z0  =  2.5  —  6/;  that  is,  for 

r0  =  2.5  ohms,  x0  =  Gohms,  z0  =  6.5  ohms, 

and  with  the  variable  conductances  as  abscissae,  the  values 
of  the  - 

output,  in  Curve  I.,  Curve  III.,  and  Curve  V.  ; 

ratio  of  potentials,  in  Curve  II.,  Curve  IV.,  and  Curve  VI.;. 

Curves  I.  and  II.  refer  to  a  non-inductive  receiver 
circuit ; 


§65] 


RESISTANCE    OF   TRANSMISSION  LINES. 


93 


Curves  III.  and  IV.  refer  to  a  receiver  circuit  of 

constant  susceptance b  =  .142 

Curves  V.  and  VI.  refer  to  a  ^receiver  circuit  of 

constant  susceptance' '  ...  .  £=—.142; 

Curves  VII.  and  VIIIvrefer  to  4  non-inductive  re- 
•  ceiver  circuit  of  a  non-inductive  line. 

In  Fig.  59,  the  output  is  shown  as  Curve  I.,  and  the 
ratio  of  potentials  as  Curve  II.,  for  the  same  line  constants, 
fora  constant  conductance,  ^  =  .0592  ohms,  and  for  variable 
susceptances,  b,  of  the  receiver  circuit. 


OUTPUT   P  AND  RATIO  OF  POTENTIAL  d  AT  RECEIVING.AND 
SENDING  END  OF  LINE  OF  IMPEDANCB  20=-2.5-3l/ 


AT  CONSTANT  IMPRESSED  E.  M.F.    E^lOOO 


#=.0592 
I    OUTPUT 
II    RATIO  OF  POTENTIALS 


SUSCEPTANCE  .OF  RECEIVER  CIRCUIT 


-3  -2  -1  0  +1          +2          +3 

Fig.  59.     Variation  of  Potential  in  Line  at  Various  Loads. 

3.)   Maximum  Efficiency. 

65.  The  output,  for  a  given  conductance,  g,  of  a  receiver 
circuit,  is  a  maximum  if  b  =  —  b0.  This,  however,  is  gen- 
erally  not  the  condition  of  maximum  efficiency. 


94  ALTERNATING-CURRENT  PHENOMENA.  [§65 

The  loss  of  energy  in  the  line  is  constant  if  the  cur- 
rent is  constant;  the  output  of  the  generator  for  a  given 
current  and  given  generator  E.M.F.  is  a  maximum  if  the  cur- 
rent is  in  phase  with  the  E.M.F.  at  the  generator  terminals. 
Hence  the  condition  of  maximum  output  at  given  loss,  or 
of  maximum  efficiency,  is  — 

tan  o>0  =  0. 
The  current  is  — 


I=E 


_  _ 

(1  +  rog  +  x0b)  —  j(x0g  —  r0b)  ' 

multiplying  numerator  and  denominator  by  (1  -f  r0g  -\-x0b) 
+  j  (x0g  —  r0b~)y  to  eliminate  the  imaginary  quantity  from 
the  denominator,  we  have  — 

/U(l  +  r0g+*ob)  -b(*0g-r0b)}  +\ 

\J  {b  (1  +  r0g  +  x0b)  +  g  (xog  -  r0b}}  ) 

(1  +  r0g  +  x0  bY  -f  (Xog  -  r0  bY 

The  current,  70,  is  in  phase  with  the  E.M.F.,  E0,  if  its 
quadrature  component  —  that  is,  the  imaginary  term  —  dis- 

appears, or 

b  (1  +  r0g+x0b)  +g(x0g-r0b)  =  0. 

This,  therefore,  is  the  condition  of  maximum  efficiency. 
Expanding,  we  have, 


Hence,  the  condition  of  maximum  efficiency  is,  that  the 
reactance  of  the  receiver  circuit  shall  be  equal,  but  of  oppo- 
site sign,  to  the  reactance  of  the  line. 

Substituting  x  =  —  x0>  we  have, 
ratio  of  KM.Fs.,  _ 

=  E_  =  _  0_  _  =  Vr2  +  *<?  . 
Eo       (r+r0}          (r+r.)    '' 

I  "  2 
nnwf»r  p  _    p  2  0-02  —       -&Q   r 

'  *•**     ~  *' 


'RESISTANCE    OF    TRANSMISSION  LINES. 


95 


and  depending  upon- the  resistance  only,  and  not  upon  the 
reactance. 

This  power  is  a  maximum  if  g  =  g0 ,  as  shown  before ; 
hence,  substituting  g  =  ga,  £  =  r0, 

ir 

E  2 
maximum  power  at  maximum  efficiency,  Pm  =       °    , 

at  a  ratio  of  potentials,        am  =     Z°    , 


or  the  same  result  as  in  §  62. 


SSEDE-.M.F.-EO-IOOO 
LME  IMPEDANCE,  Z(J=2.5  —  SJ 


CONDUCTANCE  OF  RECEIVER  CIRCUIT  Q  > 


.01  .02  .03  .04  .05  .06  .07  .08 

Fig.  60.    Load  Characteristic  of  Transmission  Line. 

In  Fig.  60  are  shown,  for  the  constants  — 
E0  =  1,000  volts, 
Z0  =  2.5  —  6y  ;    r0  =  2.5  ohms,  x0  =  6  ohms,  z0  —  6.5  ohms, 


96  ALTERNATING-CURRENT  PHENOMENA.          [§66 

and  with  the  variable  conductances,  gt  of  the  receiver  circuit 
as  abscissae,  the  — 

Output  at  maximum  efficiency,     (Curve  I.)  ; 
Volts  at  receiving  end  of  line,      (Curve  II.)  ; 

Efficiency  =  —      -  ,  (Curve  III.). 

r  4~  ro 

4.)    Control  of  Receiver  Voltage  by  Shunted  Susceptance. 

66.  By  varying  the  susceptance  of  the  receiver  circuit,, 
the  potential  at  the  receiver  terminals  is  varied  greatly. 
Therefore,  since  the  susceptance  of  the  receiver  circuit  can 
be  varied  at  will,  it  is  possible,  at  a  constant  generator 
E.M.F.,  to  adjust  the  receiver  susceptance  so  as  to  keep 
the  potential  constant  at  the  receiver  end  of  the  line,  or  to 
vary  it  in  any  desired  manner,  and  independently  of  the 
generator  potential,  within  certain  limits. 

The  ratio  of  E.M.Fs.  is  — 


r0g  +  x0b?  +  (xog  -  r0  $• 

If  at  constant  generator  potential  E0  ,  the  receiver  potential 
E  shall  be  constant, 

a  =  constant  ; 
hence, 

(1  +  r0g  +  *0  W  +  (x0g  ~  r0  by2  =  i  ; 
or,  expanding, 

b  =  -  b0 


which  is  the  value  of  the  susceptance,  b,  as  a  function  of 
the  receiver  conductance,  —  that  is,  of  the  load,  —  which  is 
required  to  yield  constant  potential,  aE0,  at  the  receiver 
circuit. 

For  increasing  g,  that  is,  for  increasing  load,  a  point  is 
reached,  where,  in  the  expression  — 


§§  67,  68]      RESISTANCE   OF   TRANSMISSION  LINES.  97 

the  term  under  the  .root  becomes  imaginary,  and  it  thus 
becomes  impossible  to  maintain  a  constant  potential,  aE0. 
Therefore,  the  maximum  output  which  can  be  transmitted 
at  potential  aE,  is  given  by  the  expression  — 


a 

hence  b  =  —  o0 ,  the  susceptance  of  receiver  circuit, 

and      g  =  —go  +  — ,  the  conductance  of  receiver  circuit ; 


a 


^-  -  g0    ,  the  output 


67.    If  a  =  1,  that  is,  if  the  voltage  at  the  receiver  cir- 
cuit equals  the  generator  potential  — 

g  =  y0  -go', 
P=a*E*(y0-go). 
If          a  =  1  when  g  =  0,       b  =  0 

when  g  >  0,        b  <  0  ; 
if  a  >  1  when  g  =  0,  or  g  >  0,  b  <  0, 

that  is,  condensance ; 
if  a  <  1  when  g  =  0,       b  >  0, 


when  g  =  -  g0  +  i/    ^  A  -  £02,  J  =  0  ; 
when  g  >  -  g0  +  J(y±\-  b0\  b  <  0, 


or,  in  other  words,  if  a  <  1,  the  phase  difference  in  the  main 
line  must  change  from  lag  to  lead  with  increasing  load. 

68.    The  value  of  a  giving  the  maximum  possible  output 
in  a  receiver  circuit,  is  determined  by  dP  I  da  =  0 ; 

expanding  :  2  a  (&-  -  g0\  -  f!f'  =  0  ; 

\a  )         a* 

hence,  y0  =  2agoj 

and  *  =      -=        l        =^-' 


98  AL  TERN  A  TING-CURRENT  PHENOMENA .  [  §  69 

the  maximum  output  is  determined  by  — 

&  ==  £>0       I  ==  <§0   j 

a 

and  is,  P  =  ^  . 

4  /- 

From  :  a  =  -2±-  =  -^_  , 

the    line    reactance,   x0,    can    be    found,   which    delivers    a 
maximum  output   into  the  receiver  circuit  at  the  ratio  of 
potentials,  a, 
and  z0  =  2  r0  a, 


xo  =  ro  V4  a1  —  1 ; 
for  a  =  1, 


If,  therefore,  the  line  impedance  equals  2  a  times  the  line 
resistance,  the  maximum  output,  P  =  E02/4:r0,  is  trans- 
mitted into  the  receiver  circuit  at  the  ratio  of  potentials,  a. 

If  z0  =  2  r0 ,  or  JTO  =  r0  V3,  the  maximum  output,  P  = 
E02/4r0,  can  be  supplied  to  the  receiver  circuit,  without 
change  of  potential  at  the  receiver  terminals. 

Obviously,  in  an  analogous  manner,  the  law  of  variation 
of  the  susceptance  of  the  receiver  circuit  can  be  found  which 
is  required  to  increase  the  receiver  voltage  proportionally  to 
the  load ;  or,  still  more  generally,  —  to  cause  any  desired 
variation  of  the  potential  at  the  receiver  circuit  indepen- 
dently of  any  variation  of  the  generator  potential,  as,  for  in- 
stance, to  keep  the  potential  of  a  receiver  circuit  constant, 
even  if  the  generator  potential  fluctuates  widely. 

69.  In  Figs.  61,  62,  and  63,  are  shown,  with  the  output, 
P  =  E* g a2,  as  abscissae,  and  a  constant  impressed  E.M.F., 
E0  =  1,000  volts,  and  a  constant  line  impedance,  Z0  = 
2.5  —  6/,  or,  r0  =  2.5  ohms,  x0  =  6  ohms,  z  =  6.5  ohms, 
the  following  values  : 


VOLTS 
1000 1— 


RATIO  OF  RECEIVER  VOLTAGE  TO  SENDER  VOLTAGE:  a  =I.O 

J    LINE  IMPEDANCE: Z0=  2.5. - 


CONSTANT  GENERATOR  POTENT 


I.    ENERGY  CURRENT 
If.    REACTIVE  CURRENT 
DI7  TOTAL  CURRENT 


IV.   CURRENT  IN  NON-INDUCTIVE  RECEIVER  CIRCUIT  WITHOUT  COMPENSATION 
"V.    POTENTIAL          "  "    '  r-  "  "  »' 


OUTPUT   IN  RECEIVER  CIRCUIT,  K 


Fig,  61.     Variation  of  Voltage  Transmission  Lines. 


RATIO  OF  RECEIVER  VOLTA3E  TO  SENDER  VOLTAGE:  a  =.7 

_LINE  IMPEDANCE:  Z0^2.5.— 


lit 


.    ENERGY  CURRENT         CONSTANT  GENERATOR  POTENTIAL  E0=IOOO 
^  REACTIVE  CURRENT 
TOTAL  CURRENT 


CURRENT  IN  NON-INDUCTIVE  CIRCUIT  WITHOUT  COMPENSATIO  1 
V.    POTENTIAL         "  "  "  "  "     " 


VOLTS 
1000 


200 
180 
160 
HO 
120 
100 
80 
60 
10 


300 
*00 
700 
600 
600 
400 
300 


\ 


OUTPUT  IN  RECEIVER  CIRCl  IT,  KILOWATTS 


30  40  50  60  70  80 

Fig.  62.     Variation  of  Voltage  Transmission  Lines. 


100 


ALTERNATING-CURRENT  PHENOMENA. 


[§69 


r    i — i — i — i — i — i — i — i — i — i — i — i — i — i — i 

RATIO  OF  RECEIVER  VOLTAGE'TO  SENDER  VOLTAQE:  a  =1  3 

'LINE  IMPEDANCE:  za=2.5.— ej — 


CONSTANT  GENERATOR  POTENTIAL  E0=IOOO 

III.  TOTAL  CURRENT  " 

IV.  CURRENT  TN  NON-INDVCTIVE  RECEIVER  CIRCUIT  WITHOUT  COMPEC 

V.  POTENTIAL 


10  20  30  40  60  60  70  80  90 

Fig.  63.     Variation  of  Voltage  Transmission  Lines. 

Energy  component  of  current,  gE,    (Curve  I.)  ; 

Reactive,  or  wattless  component  of  current,    bE,    (Curve  II.)  ; 
Total  current,  yE,    (Curve  III.)  ; 

for  the  following  conditions  : 

a  =  1.0  (Fig.  58)  ;     a=    .7  (Fig.  59)  ;     a  =  1.3  (Fig.  60). 

For  the  non-inductive  receiver  circuit  (in  dotted  lines), 
the  curve  of  E.M.F.,  E,  and  of  the  current,  I  =  gE,  are 
added  in  the  three  diagrams  for  comparison,  as  Curves  IV. 
and  V. 

As  shown,  the  output  can  be  increased  greatly,  and  the 
potential  at  the  same  time  maintained  constant,  by  the  judi- 
cious use  of  shunted  reactance,  so  that  a  much  larger  out- 
put can  be  transmitted  over  the  line  at  no  drop,  or  even  at 
a  rise,  of  potential. 


§  70]  RESISTANCE   OF   TRANSMISSION  LINES.  101 

5.)    Maximum  Rise  of  Potential  at  Receiver  Circuit. 

70.  Since,  under  certain  circumstances,  the  potential  at 
the  receiver  circuit  may  be  higher  than  at  the  g^rierator, 
it  is  of  interest  to  determine  what  is  the  maximum  value,  of 
potential,  E,  that  can  be  produced  at  the  receiver  circuit 
with  a  given  generator  potential,  E0. 

The  condition  is  that 

a  =  maximum  or  —  =  minimum; 

a'2 

that  is, 

^q/O 

- 

<*S 
substituting, 

i-  =  (1  +  r0 
and  expanding,  we  get, 


(xog  - 


—  a  value  which  is  impossible,  since  neither  r0  nor  g  can  be 
negative.  The  next  possible  value  is  g  =  0,  —  a  wattless 
circuit. 

Substituting  this  value,  we  get, 


and  by  substituting,  in 


b  -f-  b0  =  0  ; 

that  is,  the  sum  of  the  susceptances  =  0,  or  the  condition 
of  resonance  is  present. 
Substituting, 

we  have 

Vr^eT        ro       b0 


102  ALTERNATING-CURRENT  PHENOMENA.  [§71 

The  current  in  this  case  is, 


.or  tiie  «ame  as  if  the  line  resistance  were  short-circuited 
without"  any;  inductance. 

This  is  'the  condition  of  perfect  resonance,  with  current 
and  E.M.F.  in  phase. 


VOVT 
2000 

4900 
1800 
1700 
1600 
1500 
1400 
1300 
1200 
1100 
1000 
900 
800 
700 
600 
500 

N 

s 

\ 

V 

^ 

\ 

\ 

\ 

\ 

\ 

\ 

kN 

\ 

\ 

CONSTANT  IMPRESSED  E.M.F.   E0=IOOO 
"          LINE  IMPEDANCE  Z0=2.5-  6J 
\    MAXIMUM  OUTPUT  BY  COMPENSATION 
II    MAXIMUM  EFFICIENCY  BY  COMPENSATION 
III    NON-INDUCTIVE  RECEIVER  CIRCUIT 
IV   NON-INDUCTIVE  LINE  AND  NON-INDUCTIVE 
RECEIVER  CIRCUIT 

\l 

]j 

1 

—  1— 

a 

j 

-/L 

^^ 

^ 

^-^ 

*^ 

m 

X 

^ 

—II- 

N^^ 

/ 

M 

•$ 

X 

\ 

n 

1 

/ 

^ 

\ 

I/ 

/( 

\ 

400 

ff/ 

\ 

j 

800 
'200 
100 
T) 

/ 

^ 

^ 

^i\ 

^ 

,<£ 

' 

^ 

s 

^ 

^ 

HCV)C 

^ 

^ 

OUT 

PUT 

K.W 

•  — 

> 

10      20      30      40      50      60      70      80      90     100 
Fig.  64.    Efficiency  and  Output  of  Transmission  Line. 

71.  As  summary  to  this  chapter,  in  Fig.  64  are  plotted, 
for  a  constant  generator  E.M.F.,  E0  =  1000  volts,  and  a 
line  impedance,  Z0  =  2.5  —  6/,  or,  r0  =  2.5  ohms,  x0  =  6 
ohms,  z0  =  6.5  ohms  ;  and  with  the  receiver  output  as 


§71]  RESISTANCE   OF   TRANSMISSION  LINES.,  103 

abscissae   and    the    receiver   voltages    as    ordinates,   curves 
representing  — 

the  condition  of  maximum  output,"  (Curve      I.) ; 

the  condition  of  maximum  efficiency,  (Curve    II.)  ; 

the  condition  b  =  0,  or  a  rion-inductive  receiver  cir- 
cuit, (Curve  III.)  ; 

the  condition  b  =  0,  b0  =  0,  or  a  non-inductive  line  and  non- 
inductive  receiver  circuit,  or  a  non-inductive  receiver 
circuit  and  a  non-inductive  line. 

In  conclusion,  it  may  be  remarked  here  that  of  the 
sources  of  susceptance,  or  reactance, 

a  choking  coil  or  reactive  coil  corresponds  to  an  inductance ; 
a  condenser  corresponds  to  a  condensance  ; 

a  polarization  cell  corresponds  to  a  condensance  ; 

a  synchronizing  alternator  (motor  or  generator)  corresponds  to 

an  inductance  or  a  condensance,  at  will ; 
an  induction  motor  or  generator  corresponds  to  an  inductance 

or  condensance,  at  will. 

The  choking  coil  and  the  polarization  cell  are  specially 
suited  for  series  reactance,  and  the  condenser  and  syn- 
chronizer for  shunted  susceptance. 


104  ALTERNATING-CURRENT  PHENOMENA.  [§72 


CHAPTER    X. 

EFFECTIVE   RESISTANCE    AND    REACTANCE. 

72.    The  resistance  of  an  electric  circuit  is  determined  :  — 

1.)  By  direct  comparison  with  a  known  resistance  (Wheat- 
stone  bridge  method,  etc.). 

This  method  gives  what  may  be  called  the  true  ohmic 
resistance  of  the  circuit. 

2.)    By  the  ratio  : 

Volts  consumed  in  circuit 
Amperes  in  circuit 

In  an  alternating-current  circuit,  this  method  gives,  not 
the  resistance  of  the  circuit,  but  the  impedance, 


z  =      r2  +  x\ 
3.)    By  the  ratio  : 

r  =  Power  consumed  _  _     (E.M.F.)2        m 
(Current)2  Power  consumed ' 

where,  however,  the  "power"  and  the  "E.M.F."  do  not 
include  the  work  done  by  the  circuit,  and  the  counter 
E.M.Fs.  representing  it,  as,  for  instance,  in  the  case  of  the 
counter  E.M.F.  of  a  motor. 

In  alternating-current  circuits,  this  value  of  resistance  is 
the  energy  coefficient  of  the  E.M.F., 

__  Energy  component  of  E.M.F. 

Total  current 

It  is  called  the  effective  resistance  of  the  circuit,  since  it 
represents  the  effect,  or  power,  expended  by  the  circuit. 
The  energy  coefficient  of  current, 

(r_  Energy  component  of  current 

Total  E.M.F. 
is  called  the  effective  conductance  of  the  circuit. 


§  73]      EFFECTIVE  RESISTANCE   AND  REACTANCE.        105 

In  the  same  way,  the  value, 

_  Wattless  component  of  E.M.F. 
Total  current 

is  the  effective  reactance,  and  * 

riit 

,  _  Wattless  component  of  current 
Total 


is  the  effective  susceptance  of  the  circuit. 

While  the  true  ohmic  resistance  represents  the  expendi- 
ture of  energy  as  heat  inside  of  the  electric  conductor  by  a 
current  of  uniform  density,  the  effective  resistance  repre- 
sents the  total  expenditure  of  energy. 

Since,  in  an  alternating-current  circuit  in  general,  energy 
is  expended  not  only  in  the  conductor,  but  also  outside  of 
it,  through  hysteresis,  secondary  currents,  etc.,  the  effective 
resistance  frequently  differs  from  the  true  ohmic  resistance 
in  such  way  as  to  represent  a  larger  expenditure  of  energy. 

In  dealing  with  alternating-current  circuits,  it  is  necessary, 
therefore,  to  substitute  everywhere  the  values  "effective  re- 
sistance," "effective  reactance,"  "effective  conductance," 
and  "  effective  susceptance,"  to  make  the  calculation  appli- 
cable to  general  alternating-current  circuits,  such  as  ferric 
inductances,  etc. 

While  the  true  ohmic  resistance  is  a  constant  of  the 
circuit,  depending  only  upon  the  temperature,  but  not  upon 
the  E.M.F.  ,  etc.,  the  effective  resistance  and  effective  re- 
actance are,  in  general,  not  constants,  but  depend  upon 
the  E.M.F.,  current,  etc.  This  dependence  is  the  cause 
of  most  of  the  difficulties  met  in  dealing  analytically  with 
alternating-current  circuits  containing  iron. 

73.  The  foremost  sources  of  energy  loss  in  alternating- 
current  circuits,  outside  of  the  true  ohmic  resistance  loss, 
are  as  follows  : 

1.)    Molecular  friction,  as, 

a.)    Magnetic  hysteresis  ; 
If.)    Dielectric  hysteresis. 


106  ALTERNATING-CURRENT  PHENOMENA.  [§  74 

2.)    Primary  electric  currents,  as, 

a.)    Leakage  or  escape  of  current  through  the  in- 
sulation, brush  discharge ; 
b.)    Eddy  currents   in   the  conductor    or  unequal 

current  distribution. 
3.)    Secondary  or  induced  currents,  as, 

a.)    Eddy  or  Foucault  currents  in  surrounding  mag- 
netic materials; 

b.)    Eddy  or  Foucault  currents  in  surrounding  con- 
ducting materials ; 
c.)    Secondary  currents  of  mutual  inductance  in 

neighboring  circuits. 
4.)    Induced  electric  charges,  electrostatic  influence. 

While  all  these  losses  can  be  included  in  the  terms 
effective  resistance,  etc.,  only  the  magnetic  hysteresis  and 
the  eddy  currents  in  the  iron  will  form  the  subject  of  what 
follows. 

Magnetic  Hysteresis. 

74.  In  an  alternating-current  circuit  surrounded  by  iron 
or  other  magnetic  material,  energy  is  expended  outside  of 
the  conductor  in  the  iron,  by  a  kind  of  molecular  friction, 
which,  when  the  energy  is  supplied  electrically,  appears  as 
magnetic  hysteresis,  and  is  caused  by  the  cyclic  reversals  of 
magnetic  flux  in  the  iron  in  the  alternating  magnetic  field 
of  force. 

To  examine  this  phenomenon,  first  a  circuit  may  be  con- 
sidered, of  very  high  inductance,  but  negligible  true  ohmic 
resistance ;  that  is,  a  circuit  entirely  surrounded  by  iron,  as, 
for  instance,  the  primary  circuit  of  an  alternating-current 
transformer  with  open  secondary  circuit. 

The  wave  of  current  produces  in  the  iron  an  alternating 
magnetic  flux  which  induces  in  the  electric  circuit  an 
E.M.F.,  —  the  counter  E.M.F.  of  self-induction.  If  the 
ohmic  resistance  is  negligible,  the  counter  E.M.F.  equals 
the  impressed  E.M.F.  ;  hence,  if  the  impressed  E.M.F.  is 


§  75]      EFFECTIVE   RESISTANCE   AND   REACTANCE.        107 

a  sine  wave,  the  counter  E.M.F.,  and,  therefore,  the  mag- 
netic flux  which  induces  the  counter  E.M.F.  must  follow 
sine  waves  also.  The  alternating  wave  of  current  is  not  a 
.sine  wave  in  this  case,  but  i£  distorted  by  hysteresis.  It  is 
possible,  however,  to  plot  the  cfurrent  wave  in  this  case  from 
the  hysteretic  cycle  of  magnetic  flux. 

From  the  number  of  turns,  n,  of  the  electric  circuit, 
the  effective  counter  E.M.F.,  E,  and  the  frequency,  Ny 
of  the  current,  the  maximum  magnetic  flux,  <£,  is  found 
by  the  formula : 


hence,  -  _ 

"  V2  TT  n  N ' 

A  maximum  flux,  $,  and  magnetic  cross-section,  S,  give 
the  maximum  magnetic  induction,  (B  =  <£  /  5. 

If  the  magnetic  induction  varies  periodically  between 
+  (B  and  —  (B,  the  M.M.F.  varies  between  the  correspond- 
ing values  -f-  if  and  —  if,  and  describes  a  looped  curve,  the 
cycle  of  hysteresis. 

If  the  ordinates  are  given  in  lines  of  magnetic  force,  the 
abscissae  in  tens  of  ampere-turns,  then  the  area  of  the  loop 
equals  the  energy  consumed  by  hysteresis  in  ergs  per  cycle. 

From  the  hysteretic  loop  the  instantaneous  value  of 
M.M.F.  is  found,  corresponding  to  an  instantaneous  value 
of  magnetic  flux,  that  is,  of  induced  E.M.F.  ;  and  from  the 
M.M.F.,  if,  in  ampere-turns  per  unit  length  of  magnetic  cir- 
cuit, the  length,  /,  of  the  magnetic  circuit,  and  the  number  of 
turns,  n,  of  the  electric  circuit,  are  found  the  instantaneous 
values  of  current,  /,  corresponding  to  a  M.M.F.,  if,  that  is, 
.as  magnetic  induction  (B,  and  thus  induced  E.M.F.  e,  as  : 

n 

75.  In  Fig.  65,  four  magnetic  cycles  are  plotted,  with 
maximum  values  of  magnetic  inductions,  (B  =  2,000,  6,000, 
10,000,  and  16,000,  and  corresponding  maximum  M.M.Fs., 


108 


AL  TERNA  TING-CURRENT  PHENOMENA. 


75 


SF  =  1.8,  2.8,  4.3,  20.0.  They  show  the  well-known  hys- 
teretic  loop,  which  becomes  pointed  when  magnetic  satu- 
ration is  approached. 

These  magnetic  cycles  correspond  to  average  good  sheet 
iron  or  sheet  steel,  having  a  hysteretic  coefficient,  rj  =  .0033,. 
and  are  given  with  ampere-turns  per  cm  as  abscissae,  and 
kilo-lines  of  magnetic  force  as  ordinates. 


Fig.  65.    Hysteretic  Cycle  of  Sheet  Iron. 

In  Figs.  66,  67,  68,  and  69,  the  sine  curve  of  magnetic 
induction  as  derived  from  the  induced  E.M.F.  is  plotted  in 
dotted  lines.  For  the  different  values  of  magnetic  induction 
of  this  sine  curve,  the  corresponding  values  of  M.M.F.,  hence 
of  current,  are  taken  from  Fig.  66,  and  plotted,  giving  thus 
the  exciting  current  required  to  produce  the  sine  wave  of 
magnetism ;  that  is,  the  wave  of  current  which  a  sine  wave 
of  impressed  E.M.F.  will  send  through  the  circuit. 


§  75]      EFFECTIVE  RESISTANCE   AND   REACTANCE.         109 

As  shown  in  Figs.  66,  67,  68,  and  69,  these  waves  of 
alternating  current  are  not  sine  waves,  but  are  distorted  by 
the  superposition  of  higher .  harmonics,  and  are  complex 
harmonic  waves.  They  reach  their  maximum  value  at  the 
same  time  with  the"  maximum  of  magnetism,  that  is,  90° 


=  =  2000 


3-1.8 


\ 


CB-6000 


3  "2.9 


X 


7 


Figs.  66  and  67.     Distortion  of  Current  Wave  by  Hysteresis. 


ahead  of  the  maximum  induced  E.M.F.,  and  hence  about 
90°  behind  the  maximum  impressed  E.M.F.,  but  pass  the 
zero  line  considerably  ahead  of  the  zero  value  of  magnet- 
ism, or  42°,  52°,  50°,  and  41°,  respectively. 

The  general  character  of  .these  current  waves  is,  that  the 
maximum  point  of  the  wave  coincides  in  time  with  the  max- 


110 


ALTERNATING-CURRENT  PHENOMENA. 


75 


imum  point  of  the  sine  wave  of  magnetism  ;  but  the  current 
wave  is  bulged  out  greatly  at  the  rising,  and  hollowed  in  at 
the  decreasing,  side.  With  increasing  magnetization,  the 
maximum  of  the  current  wave  becomes  more  pointed,  as 
shown  by  the  curve  of  Fig.  68,  for  &  =  10,000  ;  and  at  still 


/\\ 


NX 


f\ 


16000 


A 


20 


\ 


\ 


13 


\J 


\J 


Figs.  68  and  69.     Distortion  of  Current  Wave  by  //ysferes/s. 

higher  saturation  a  peak  is  formed  at  the  maximum  point, 
as  in  the  curve  of  Fig.  69,  for  (B  =  16,000.  This  is  the 
case  when  the  curve  of  magnetization  remains  within  the 
range  of  magnetic  saturation,  since  in  the  proximity  of  satu- 
ration the  current  near  the  maximum  point  of  magnetization 
has  to  rise  abnormally  to  cause  even  a  small  increase  of 
magnetization. 


§§76,77]  EFFECTIVE  RESISTANCE  AND  REACTANCE.    Ill 

The  four  curves,  Figs.  66,  67,  68,  and  C9,  are  not  drawn 
to  the  same  scale.  The  maximum  values  of  M.M.F.,  cor- 
responding to  the  maximimT  values  of  magnetic  induction, 
<B  =  2,000,  6,000,  10,000,  ajid  16,000  lines  of  force  per 
cm2,  are  $  =1.8,  £8,4.3,  and  20.0  ampere-turns  per  cm. 
In  the  different  diagrams  these  are  represented  in  the  ratio 
of  8:6:4:1,  in  order  to  bring  the  curves  of  current  to 
approximately  the  same  height. 

The  M.M.F.,  in  C.G.S.  units,  is 

*=*  =  1-257  F. 


76.  The  distortion  of  the  wave  of  magnetizing  current 
is  as  large  as  shown  here  only  in  an  iron-closed  magnetic 
circuit  expending  energy  by  hysteresis  only,  as  in  an  iron- 
clad transformer  on  open  secondary  circuit.      As  soon  as  the 
circuit  expends  energy  in  any  other  way,  as  in  resistance,  or 
by  mutual  inductance,  or  if  an  air-gap  is  introduced  in  the 
magnetic  circuit,  the  distortion  of  the  current  wave  rapidly 
decreases  and  practically  disappears,  and  the  current  becomes 
more  sinusoidal.     That  is,  while  the  distorting  component 
remains  the  same,  the  sinusoidal  component  of  the  current 
greatly  increases,  and  obscures  the  distortion.      For  example, 
in  Figs.  70  and  71,  two  waves  are  shown,  corresponding  in 
magnetization   to   the   curve   of   Fig.   67,  as   the  one  most 
distorted.     The  curve  in  Fig.  70  is  the  current  wave  of  a 
transformer  at  -^  load.     At  higher  loads  the  distortion  is 
correspondingly  still   less.      The   curve  *of   Fig.   71    is    the 
exciting  current  of  a  magnetic  circuit  containing  an  air-gap 
whose  length  equals  3^  the  length  of  the  magnetic  circuit. 
These  two  curves  are  drawn  to  ^  the  size  of  the  curve  in 
Fig.  67.     As  shown,  both  curves  are  practically  sine  waves. 

77.  The  distorted  wave  of  current  can  be  resolved  into 
two  components  :  A  true  sine  wave  of  equal  effective  intensity 
and  equal  power  to  tlie  distorted  wave,  called  the  equivalent 


112 


ALTERNATING-CURRENT  PHENOMENA. 


[§77 


sine  wave,  and  a  wattless  higher  harmonic,  consisting  chiefly 
of  a  term  of  triple  frequency. 

In  Figs.  66  to  71  are  shown,  in  full  lines,  the  equiva- 


Figs.  70  and  71.    Distortion  of  Current  Wave  by  Hysteresis. 

lent  sine  waves  and  the  wattless  complex  higher  harmonics, 
which  together  form  the  distorted  current  wave.  The 
equivalent  sine  wave  of  M.M.F.  or  of  current,  in  Figs.  66 
to  69,  leads  the  magnetism  by  34°,  44°,  38°,  and  15°.5, 


§  78]       EFFECTIVE   RESISTANCE   AND   REACTANCE.         113 

respectively.  In  Fig.  71  the  equivalent  sine  wave  almost 
coincides  with  the  distorted  curve,  and  leads  the  magnetism 
by  only  9°.  '*;'* 

It  is  interesting  to  nor*e,  that  even  in  the  greatly  dis- 
torted curves  of  Figs.  66  ton^S,  the  maximum  value  of  the 
equivalent  sine  wave  is  nearly  the  same  as  the  maximum 
value  of  the  original  distorted  wave  of  M.M.F.,  so  long  as 
magnetic  saturation  is  not  approached,  being  1.8,  2.9,  and 
4.2,  respectively,  against  1.8,  2.8,  and  4.3,  the  maximum 
values  of  the  distorted  curve.  Since,  by  the  definition,  the 
effective  value  of  the  equivalent  sine  wave  is  the  same  as 
that  of  the  distorted  wave,  it  follows,  that  the  distorted 
wave  of  exciting  current  shares  with  the  sine  wave  the 
feature,  that  the  maximum  value  and  the  effective  value 
have  the  ratio  of  V2  -f-  1.  Hence,  below  saturation,  the' 
maximum  value  of  the  distorted  curve  can  be  calculated 
from  the  effective  value  —  which  is  given  by  the  reading 
of  an  electro-dynamometer  —  by  using  the  same  ratio  that 
applies  to  a  true  sine  wave,  and  the  magnetic  characteris- 
tic can  thus  be  determined  by  means  of  alternating  cur- 
rents, with  sufficient  exactness,  by  the  electro-dynamometer 
method. 

78.  In  Fig.  72  is  shown  the  true  magnetic  character- 
istic of  a  sample  of  good  average  sheet  iron,  as  found  by 
the  method  of  slow  reversals  with  the  magnetometer  ;  for 
comparison  there  is  shown  in  dotted  lines  the  same  char- 
acteristic, as  determined  with  alternating  currents  by  the 
electro-dynamometer,  with  ampere-turns  per  cm  as  ordi- 
nates,  and  magnetic  inductions  as  abscissas.  As  repre- 
sented, the  two  curves  practically  coincide  up  to  (B  =  10,000 
to  14,000  ;  that  is,  up  to  the  highest  inductions  practicable 
in  alternating-current  apparatus.  For  higher  saturations, 
the  curves  rapidly  diverge,  and  the  electro-dynamometer 
curve  shows  comparatively  small  M.M.Fs.  producing  appar- 
ently very  high  magnetizations. 


11-4 


AL  TERNA  TING-CURRENT  PHENOMENA. 


[§78 


The  same  Fig.  72  gives  the  curve  of  hysteretic  loss,  in 
ergs  per  cm3  and  cycle,  as  ordinates,  and  magnetic  induc- 
tions as  abscissae. 


20 
19- 
18- 
17- 
16- 
15- 
14- 
13- 
12- 
11- 


5=1,000  2,000  3,000  4,000  5,000  0,000  7,000  8,000   9,000  10,000  11,000  12,000 13,000^4,000 15,000 16,000  17.0UO 
Fig.  72.    Magnetization  and  Hysteresis  Curve. 

The  electro-dynamometer  method  of  determining  the 
magnetic  characteristic  is  preferable  for  use  with  alter- 
nating-current apparatus,  since  it  is  not  affected  by  the 
phenomenon  of  magnetic  "creeping,"  which,  especially  at 


§  79]      EFFECTIVE   RESISTANCE   AND   REACTANCE.         115 

low  densities,  brings  in  the  magnetometer  tests  the  magnet- 
ism usually  very  much  .higher,  or  the  M.M.F.  lower,  than 
found  in  practice  in  alternating-current  apparatus. 

So  far  as  current  strength  and  energy  consumption  are 
concerned,  the  distorted  wave*  can  be  replaced  by  the  equi- 
valent sine  wave,  and  the  higher  harmonic  neglected. 

All  the  measurements  of  alternating  currents,  with  the 
single  exception  of  instantaneous  readings,  yield  the  equiv- 
alent sine  wave  only,  and  suppress  the  higher  harmonic  ; 
since  all  measuring  instruments  give  either  the  mean  square 
of  the  current  wave,  or  the  mean  product  of  instantaneous 
values  of  current  and  E.M.F.,  which,  by  definition,  are  the 
same  in  the  equivalent  sine  wave  as  in  the  distorted  wave. 

Hence,  in  all  practical  applications,  it  is  permissible  to 
neglect  the  higher  harmonic  altogether,  and  replace  the  dis- 
torted wave  by  its  equivalent  sine  wave,  keeping  in  mind, 
however,  the  existence  of  a  higher  harmonic  as  a  possible 
disturbing  factor  which  may  become  noticeable  in  those  very 
rare  cases  where  the  frequency  of  the  higher  harmonic  is 
near  the  frequency  of  resonance  of  the  circuit. 

79.  The  equivalent  sine  wave  of  exciting  current -leads 
the  sine  wave  of  magnetism  by  an  angle  a,  which  is  called 
the  angle  of  hysteretic  advance  of  phase.  Hence  the  cur- 
rent lags  behind  the  E.M.F  by  ^Y  90°  —  a,  and  the  power 
is  therefore, 

P  =  IE  cos  (90°  —  a)  ==  IE  sin  a. 

Thus  the  exciting  current,  /,  consists  of  an  energy  com- 
ponent, /  sin  a,  which  is  called  the  hysteretic  energy  current, 
and  a  wattless  component,  /  cos  a,  which  is  called  the  mag- 
netising current.  Or,  conversely,  the  E.M.F.  consists  of  an 
energy  component,  E  sin  a,  the  hysteretic  energy  E.M.F., 
and  a  wattless  component,  E  cos  a,  the  E.M.F.  of  self- 
induction. 

Denoting  the  absolute  value   of   the   impedance  of  the 


116  ALTERNATING-CURRENT  PHENOMENA.  [§  8O 

circuit,  E I  It  by  z,  —  where  z  is  determined  by  the  mag- 
netic characteristic  of  the  iron,  and  the  shape  of  the 
magnetic  and  electric  circuits,  —  the  impedance  is  repre- 
sented, in  phase  and  intensity,  by  the  symbolic  expression, 

Z  =  r  —  jx  =  z  sin  a  —  jz  cos  a ; 
and  the  admittance  by, 

Y  =  g  +  j  b  =  -  sin  a  +  j  -  cos  a  =  y  sin  a  +  jy  cos  a. 

z  z 

The  quantities,  z,  r,  x,  and  y,  g,  b,  are,  however,  not 
constants  as  in  the  case  of  the  circuit  without  iron,  but 
depend  upon  the  intensity  of  magnetization,  (B,  —  that  is, 
upon  the  E.M.F. 

This  dependence  complicates  the  investigation  of  circuits 
containing  iron. 

In  a  circuit  entirely  inclosed  by  iron,  a  is  quite  consider- 
able, ranging  from  30°  to  50°  for  values  below  saturation. 
Hence,  even  with  negligible  true  ohmic  resistance,  no  gfeat 
lag  can  be  produced  in  ironclad  alternating-current  circuits. 

80.  The  loss  of  energy  by  hysteresis  due  to  molecular 
friction  is,  with  sufficient  exactness,  proportional  to  the 
1.6th  power  of  magnetic  induction  (R.  Hence  it  can  be  ex- 
pressed by  the  formula : 

WM±1&* 

where  — 

WH  =  loss  of  energy  per  cycle,  in  ergs  or  (C.G.S.)  units  (=  10~7 

Joules)  per  cm8, 

<B  =  maximum  magnetic  induction,  in  lines  of  force  per  cm2,  and 
•YJ  =  the  coefficient  of  hysteresis. 

This  I  found  to  vary  in  iron  from  .00124  to  .0055.  As  a 
fair  mean,  .0033  can  be  accepted  for  good  average  annealed 
sheet  iron  or  sheet  steel.  In  gray  cast  iron,  rj  averages 
.013  ;  it  varies  from  .0032  to  .028  in  cast  steel,  according 
to  the  chemical  or  physical  constitution ;  and  reaches  values 
as  high  as  .08  in  hardened  steel  (tungsten  and  manganese 


§  80]        EFFECTIVE   RESISTANCE   AND   REACTANCE.       117 

steel).  Soft  nickel  and  cobalt  have  about  the  same  co- 
efficient of  hysteresis  as  gray  cast  iron  ;  in  magnetite  I 
found  77  =  .023. 

In  the  curves  of  Fig.  62  1^69,  r,  =  .0033. 

At  the  frequency,  TV,  the  loss  of  power  in  the  volume,  V, 
is,  by  this  formula,  — 

P  =  rj  N  F&1-6  10  -  7  watts 
=  77  N  V    -^Y'6 10  -7  watts, 


where  5  is  the  cross-section  of  the  total  magnetic  flux,  <£. 

The    maximum    magnetic     flux,    <£,    depends    upon    the 
counter  E.M.F.  of  self-induction, 

E  =  V2  TT  Nn  $>  10  - 8, 
or  * 


V2  TT  Nn 

where  n  —  number  of  turns  of  the  electric  circuit. 

Substituting   this    in   the    value   of   the  power,   P,    and 
canceling,  we  get,  — 

F  1>6                 l/"105'8  7?  1.6 

P  =   77 ^M =    58  77  ^- 


,  where  ^ 


or,  substituting  77  =  .0033,  we  have  A  =  191.4     19    19  5 

o   '  /z 

or,  substituting  F=  SL,   where  L  =  length  of  magnetic  circuit, 
A  =       ryZlO5-8        =  58  r,  ZIP3  =  x  Z 


and  ^  =  58  9  ^  ]'6  Z  103  =  191'4 

- 


In  Figs.  73,  74,  and  75,  is  shown  a  curve  of  hysteretic 
loss,  with  the  loss  of  power  as  abscissae,  and 

in  curve  73,  with  the  E.M.F.,  E,  as  ordinates,  for  Z  =  1,  S  =  1, 
N=  100,  and  n  =  100  ; 


118 


AL  TERNA  TING-CURRENT  PHENOMENA. 


[§80 


9000 

/ 

' 

/ 

/ 

RELATI 

ON 

ET\ 

VEEN  E 

AND 

P 

/ 

6000 

F 

OR 

L-1 

,  s= 

-1, 

4-1 

00 

71  = 

10C 

/ 

7 

/ 

/ 

/ 

6000 
4000 

LJ 

* 

/ 

2 
i 

/ 

Q- 

/< 

^ 

/ 

/ 

/ 

1000 

S 

P 

x 

/r 

,X 

lx 

.  ' 

^ 

:.M 

F. 

Fig.  73.    Hysteresis  Loss  as  Function  of  E,  M.  F. 


1.3 


1.0 


ETW 


ILA 


TION. 


EEN 


n  A 


ND 


LT1,  S=1    N 


100.  E=100 


=  NUMBER  OF  1 


URNS 


50       100      150      200      250      300      350      400 

Fig.  74.    Hysteresis  Loss  as  Function  of  Number  of  Turns. 


§81]     EFFECTIVE  RESISTANCE  AND   REACTANCE, 


RELATION  BETWEEN   N  AND  P 
FOR  S-l,   L  =  l,  U  =  IOO,   E  =  IOO 


)0  200  300 

75.     Hysteresis  Loss  as  Function  of  Cycles. 

in  curve  74,  with  the  number  of  turns  as  ordinates,  for 
Z  =  1,  5=1,^=100,  and  .£  =  100; 

in  curve  75,  with  the  frequency,  JVt  or  the  cross-section,  S9 
as  ordinates,  for  L  =  1,  n  =  100,  and  E  =  100. 

As  shown,  the  hysteretic  loss  is  proportional  to  the  1.6th 
power  of  the  E.M.F.,  inversely  proportional  to  the  1.6th 
power  of  the  number  of  turns,  and  inversely  proportional  to 
the  .6th  power  of  frequency,  and  of  cross-section. 

81.  If  g  =  effective  conductance,  the  energy  compo- 
nent of  a  current  is  /  =  Eg,  and  the  energy  consumed  in 
a  conductance,  gt  is  P  =  IE  =  E^g. 

Since,  however  : 

Z7- 1.6  EM.6 

7~»  A     «*•*  1_  A    J-*  Z7*2 

^•6    = 

or 


AT"  6    '  iicivv^      - 

58  77 ZIP3 


From  this  we  have  the  following  deduction : 


120 


ALTERNATING-CURRENT  PHENOMENA. 


[§81 


The  effective  conductance  due  to  magnetic  hysteresis  is 
proportional  to  the  coefficient  of  hysteresis,  ??,  and  to  the  length 
of  the  magnetic  circuit,  L,  and  inversely  proportional  to  the 
Jj,th  power  of  the  E.M.F.,  to  the  .6th  power  of  the  frequency, 
N,  and  of  the  cross-section  of  the  magnetic  circuit,  S,  and  to 
the  1.6th  power  of  the  number  of  turns,  n. 

Hence,  the  effective  hysteretic  conductance  increases 
with  decreasing  E.M.F.,  and  decreases  with  increasing 


RELATION  BETWEEN     flfAND  E 

FOR  L  =  l,'  N=|00.  S—\,  n= 


50  100  150  200  250  300  350 

Fig.  76.    Hysteresis  Conductance  as  Function  of  E.M.F. 

E.M.F. ;  it  varies,  however,  much  slower  than  the  E.M.F., 
so  that,  if  the  hysteretic  conductance  represents  only  a  part 
of  the  total  energy  consumption,  it  can,  within  a  limited 
range  of  variation  —  as,  for  instance,  in  constant  potential 
transformers  —  be  assumed  as  constant  without  serious 
error. 

In  Figs.  76,  77,  and  78,  the  hysteretic  conductance,  g,  is 
plotted,  for  L  =  1,  E  =  100,  N=  100,  5  =  1,  and  n  -  100, 
respectively,  with  the  conductance,  g,  as  ordinates,  and  with 


§81]     EFFECTIVE  RESISTANCE   AND   REACTANCE.         121 


RELATION  BETWEEN    Q  AND  N 

FOR  L-l.  E  =  IOO.  8-1,  n-\OO 


50  100  150  200  250  300  350  .400 

Fig.  77.     Hysteresis  Conductance  as  Function  of  Cycles. 


1 

j 

R 

.LA' 

ION  BE' 

'WE 

EN 

IAN  Dfl 

FOF 

L= 

1,  E 

=  1( 

30, 

N^ 

100 

,  8  = 

-1. 

\ 

0 

\ 

06 

\ 

\ 

^v 

V 

\ 

S. 

=  NL 

^ 

MB 

^= 

:R  o 

•  —  * 

FT 

==^ 

JRN 

n 

50                100                150               200                250               300                350               400  '. 
Fig.  78.     Hysteresis  Conductance  as  Function  of  Number  of  Turns. 

122  ALTERNATING-CURRENT  PHENOMENA.          [§82 

E  as  abscissas  'in  Curve  76. 
-A^as  abscissae  in  Curve  77. 
n  as  abscissae  in  Curve  78. 

As  shown,  a  variation  in  the  E.M.F.  of  50  per  cent 
causes  a  variation  in  g  of  only  14  per  cent,  while  a  varia- 
tion in  N  or  5  by  50  per  cent  causes  a  variation  in  g  of  21 
per  cent. 

If  (R  =  magnetic  reluctance  of  a  circuit,  JFA  =  maximum 
M.M.F.,  /  =  effective  current,  since  /  V2  =  maximum  cur- 
rent, the  magnetic  flux, 


(R  (R 

Substituting  this  in  the  equation  of  the  counter  E.M.F.  of 
self-induction, 

£  =-V2 

we  have  ^2. 


(R 
hence,  the  absolute  admittance  of  the  circuit  is 

(R1°8 


E        I-K^N         N' 


108 
where  a  = ,  a  constant. 

2-n-n2 

Therefore,  the  absolute  admittance,  y,  of  a  circuit  of  neg- 
ligible resistance  is  proportional  to  the  magnetic  rehictance,  (R, 
and  inversely  proportional  to  the  frequency,  N,  and  to  the 
square  of  the  number  of  turns,  n. 

82.  In  a  circuit  containing  iron,  the  reluctance,  (R,  varjes 
with  the  magnetization;  that  is,  with  the  E.M.F.  Hence 
the  admittance  of  such  a  circuit  is  not  a  constant,  but  is 
also  variable. 

In  an  ironclad  electric  circuit,  —  that  is,  a  circuit  whose 
magnetic  field  exists  entirely  within  iron,  such  as  the  mag- 
netic circuit  of  a  well-designed  alternating-current  trans- 


§  82]       EFFECTIVE   RESISTANCE   AND   REACTANCE.        123 

former,  —  (R  is  the  reluctance  of  the  iron  circuit.     Hence, 
if  /A  =  permeability,  since  — 


and  <5^  =  LF=--LW  =  M.M.F., 

4t  7T 

3>  —  ,5"  (B  =  ^  £  5C  =  magnetic  flux, 
and  (R  =  ; 

4:  TT  (JL  S 

substituting  this  value  in  the  equation  of  the  admittance, 

(R  IP8  Z  IP9  z 

y  =  * s-Tr*  we  have  zr^r-7 


where  *  =  M^  =  ^^ 


Therefore,  in  art  ironclad  circuit,  the  absolute  admittance, 
y,  is  inversely  proportional  to  the  frequency,  N,  to  the  perme- 
ability, /A,  to  the  cross-section,  S,  and  to  the  square  of  the 
number  of  turns,  n  ;  and  directly  proportional  to  the  length 
of  the  magnetic  circuit,  L. 

A 

The  conductance  is         g  = — -  ; 

and  the  admittance,  y  =  -?—  ; 

hence,  the  angle  of  hysteretic  advance  is 

Sin  a  =  -£-  =  ^ ; 

y        zE* 

or,  substituting  for  A  and  z  (p.  117), 


NA  rjZlO5'8  S 

Sin  a  —  a  -       —  '-  -       -  —  , 

£•*       2-87r1-661'6/2L6      Z109   ' 


or,  substituting 

E  =  2 

we  have        sin  a  =  — 


124  AL  TERNA  TING-CURRENT  PHENOMENA.          [  §  83 

which  is  independent  of  frequency,  number  of  turns,  and 
shape  and  size  of  the  magnetic  and  electric  circuit. 

Therefore,  in  an  ironclad  inductance,  the  angle  of  hysteretic 
advance,  a,  depends  upon  the  magnetic  constants,  permeability 
and  coefficient  of  hysteresis,  and  upon  the  maximum  magnetic 
induction,  but  is  entirely  independent  of  the  frequency,  of  the 
shape  and  other  conditions  of  the  magnetic  and  electric  circtiit ; 
and,  therefore,  all  ironclad  magnetic  circuits  constructed  of  the 
same  quality  of  iron  and  using  the  same  magnetic  density, 
give  the  same  angle  of  hysteretic  advance. 

The  angle  of  hysteretic  advance,  a,  in  a  closed  circuit 
transformer,  depends  upon  the  quality  of  the  iron,  and  upon 
the  magnetic  density  only. 

The  sine  of  the  angle  of  hysteretic  advance  equals  Jf,  times 
the  product  of  the  permeability  and  coefficient  of  hysteresis, 
divided  by  the  .Jj?h  power  of  tJie  magnetic  density. 

83.  If  the  magnetic  circuit  is  not  entirely  ironclad, 
and  the  magnetic  structure  contains  air-gaps,  the  total  re- 
luctance is  the  sum  of  the  iron  reluctance  and  of  the  air 
reluctance,  or 

(R  =  (R .  _|_  (Ro  . 

hence  the  admittance  is 


Therefore,  in  a  circuit  containing  iron,  the  admittance  is 
the  sum  of  the  admittance  due  to  the  iron  part  of  the  circuit, 
y{  =  a  I  N(S(.iy  and  of  the  admittance  due  to  the  air  part  of  tJie 
circuit,  ya  =  a  I  N(&a,  if  the  iron  and  the  air  are  in  series  in 
the  magnetic  circuit. 

The  conductance,  g,  represents  the  loss  of  energy  in 
the  iron,  and,  since  air  has  no  magnetic  hysteresis,  is  not 
changed  by  the  introduction  of  an  air-gap.  Hence  the 
angle  of  hysteretic  advance  of  phase  is 


sin  a  = 


§  84]      EFFECTIVE   RESISTANCE   AND   REACTANCE.         125 

and  a  maximum, gjyiy  for  the  ironclad  circuit,  but  decreases 
with  increasing  width  of  the  air-gap.  The  introduction  of 
the  air-gap  of  reluctance,  (Ra,  decreases  sin  a  in  the  ratio, 


In  the  range  of  practical  application,  from  (B  =  2,000  to 
(B  =  12,000,  the  permeability  of  iron  varies  between  900 
and  2,000  approximately,  while  sin  a  in  an  ironclad  circuit 
varies  in  this  range  from  .51  to  .69.  In  air,  /*  =  1. 

If,  consequently,  one  per*  cent  of  the  length  of  the  iron 
is  replaced  by  an  air-gap,  the  total  reluctance  only  varies 
in  the  proportion  of  1^  to  l^o*  °r  about  6  per  cent,  that  is, 
remains  practically  constant ;  while  the  angle  of  hysteretic 
advance  varies  from  sin  a  =  .035  to  sin  a  =  .064.  Thus  g  is 
negligible  compared  with  b,  and  b  is  practically  equal  to  y. 

Therefore,  in  an  electric  circuit  containing  iron,  but 
forming  an  open  magnetic  circuit  whose  air-gap  is  not  less 
than  T^o  the  length  of  the  iron,  the  susceptance  is  practi- 
cally constant  and  equal  to  the  admittance,  so  long  as 
saturation  is  not  yet  approached,  or, 

t       G\.a  N 

b  —  — -  ,   or  :  x  =  —  . 

N  (Ra 

The  angle  of  hysteretic  advance  is  small,  below  4°,  and  the 
hysteretic  conductance  is, 

A 

g  ~  EANA  ' 

The  current  wave  is  practically  a  sine  wave. 

As  an  instance,  in  Fig.  71,  Curve  II.,  the  current  curve 
of  a  circuit  is  shown,  containing  an  air-gap  of  only  ?^  of 
the  length  of  the  iron,  giving  a  current  wave  much  resem- 
bling the  sine  shape,  with  an  hysteretic  advance  of  9°. 

84.    To    determine  the  electric  constants   of    a   circuit 
containing  iron,  we  shall  proceed  in  the  following  way : 
Let- 

E  =  counter  E.M.F.  of  self-induction  ; 


126  ALTERNATING-CURRENT  PHENOMENA.          [§84 

then  from  the  equation, 

E=  V27r«yV<I>10-8, 
where, 

JV  =  frequency, 

n   =  number  of  turns, 

we  get  the  magnetism,  <£,  and  by  means  of  the  magnetic  cross 
section,  S,  the  maximum  magnetic  induction  :  (B  =  <£  /  .S. 

From  (&,  we  get,  by  means  of  the  magnetic  characteristic 
of  the  iron,  the  M.M.F.,  =  SF  ampere-turns  per  cm  length, 
where 

SF^ae, 

4  IT 

if  OC  =  M.M.F.  in  C.G.S.  units. 
Hence, 

if   Li  —  length    of   iron    circuit,   9^  =  Z,  $  =  ampere-turns    re- 
quired in  the  iron  ; 

if  La  =  length  of  air  circuit,  SFa  =  -  -  —  =  ampere-turns  re- 

,  .     ^  4  TT 

quired  m  the  air  ; 

hence,  &=  &t  -\-  $a  =  total  ampere-turns,  maximum  value, 
and  $  I  V2  =  effective  value.     The  exciting  current  is 


and  the  absolute  admittance, 

y  =  VPTT2  =  L  . 

±L 

If  SFt-  is  not  negligible  as  compared  with  CFa,  this  admit- 
tance, y,  is  variable  with  the  E.M.F.,  E. 

If  — 

V  =  volume  of  iron, 

rj   =  coefficient  of  hysteresis, 

the  loss  of  energy  by  hysteresis  due  to  molecular  magnetic 
friction  is, 


hence  the  hysteretic  conductance  is  g  =  W  '  j  E?,  and  vari- 
able with  the  E.M.F.,  E. 


§85]      EFFECTIVE   RESISTANCE   AND   REACTANCE.         127 

The  angle  of  hysteretic  advance  is,  — 
sin  a  =  £- ; 

,  y  _____ 

the  susceptance,  b  =  Vy2'  —  g* ; 


the  effective  resistance, 


. 


and  the  reactance,  ,  x  =  — 

y2 

85.  'As  conclusions,  we  derive  from  this  chapter  the 
following  :  — 

1.)  Iii  an  alternating-current  circuit  surrounded  by  iron, 
the  current  produced  by  a  sine  wave  of  E.M.F.  is  not  a 
true  sine  wave,  but  is  distorted  by  hysteresis. 

2.)  This  distortion  is  excessive  only  with  a  closed  mag- 
netic circuit  transferring  no  energy  into  a  secondary  circuit 
by  mutual  inductance. 

3.)  The  distorted  wave  of  current  can  be  replaced  by 
the  equivalent  sine  wave  —  that  is  a  sine  wave  of  equal  effec- 
tive intensity  and  equal  power  —  and  the  superposed  higher 
harmonic,  consisting  mainly  of  a  term  of  triple  frequency, 
may  be  neglected  except  in  resonating  circuits. 

4.)  Below  saturation,  the  distorted  curve  of  current  and 
its  equivalent  sine  wave  have  approximately  the  same  max- 
imum value. 

5.)  The  angle  of  hysteretic  advance,  —  that  is,  the  phase 
difference  between  the  magnetic  flux  and  equivalent  sine 
wave  of  M.M.F.,  —  is  a  maximum  for  the  closed  magnetic 
circuit,  and  depends  there  only  upon  the  magnetic  constants 
of  the  iron,  upon  the  permeability,  /*,  the  coefficient  of  hys- 
teresis, -r),  and  the  maximum  magnetic  induction,  as  shown  in 

the  equation,  4 

sin  a  =  —  f—  L  . 

(B-4 

6.)  The  effect  of  hysteresis  can  be  represented  by  an 
admittance,  Y  =  g  +j  b,  or  an  impedance,  Z  =  r  —  jx. 

7.)  The  hysteretic  admittance,  or  impedance,  varies  with 
the  magnetic  induction;  that  is,  with  the  E.M.F.,  etc. 


128  ALTERNATING-CURRENT  PHENOMENA.          [§85 

8.)  The  hysteretic  conductance,  £-,  is  proportional  to  the 
coefficient  of  hysteresis,  17,  and  to  the  length  of  the  magnetic 
circuit,  L,  inversely  proportional  to  the  .4th  power  of  the 
E.M.F.,  E,  to  the  .6th  power  of  frequency,  N,  and  of  the 
cross-section  of  the  magnetic  circuit,  S,  and  to  the  1.6th 
power  of  the  number  of  turns  of  the  electric  circuit,  n,  as 

expressed  in  the  equation, 

58  r,  L  108 

g    =  £AJV-GS-6n1-6  ' 

9.)    The  absolute  value  of  hysteretic  admittance,  — 


is  proportional  to  the  magnetic  reluctance  :  (R  =  <ftt  -f  &a  » 
and  inversely  proportional  to  the  frequency,  N,  and  to  the 
square  of  the  number  of  turns,  n,  as  expressed  in  the 

ec*uation>  _(«,     «    io' 


10.)  In  an  ironclad  circuit,  the  absolute  value  of  admit- 
tance is  proportional  to  the  length  of  the  magnetic  circuit, 
and  inversely  proportional  to  cross-section,  S,  frequency,  N, 
permeability,  /x,  and  square  of  'the  number  of  turns,  n,  or 

127  L  106 


11.)  In  an  open  magnetic  circuit,  the  conductance,  g,  is 
the  same  as  in  a  closed  magnetic  circuit  of  the  same  iron  part. 

12.)  In  an  open  magnetic  circuit,  the  admittance,  y,  is 
practically  constant,  if  the  length  of  the  air-gap  is  at  least 
T£<y  of  the  length  of  the  magnetic  circuit,  and  saturation  be 
not  approached. 

13.)  In  a  closed  magnetic  circuit,  conductance,  suscep- 
tance,  and  admittance  can  be  assumed  as  constant  through 
a  limited  range  only. 

14.)  From  the  shape  and  the  dimensions  of  the  circuits, 
and  the  magnetic  constants  of  the  iron,  all  the  electric  con- 
stants, g,  b,  y  \  r*  x,  zt  can  be  calculated. 


§86]  FOUCAULT  OR   EDDY  CURRENTS.  129 


CHAPTER    XI. 

FOUCAULT  OR  EDDY  CURRENTS. 

86.  While  magnetic  hysteresis  or  molecular  friction  is 
a  magnetic  phenomenon,  eddy  currents  are  rather  an  elec- 
trical phenomenon.  When  iron  passes  through  a  magnetic 
field,  a  loss  of  energy  is  caused  by  hysteresis,  which  loss, 
however,  does  not  react  magnetically  upon  the  field.  When 
cutting  an  electric  conductor,  the  magnetic  field  induces  a 
current  therein.  The  M.M.F.  of  this  current  reacts  upon 
and  affects  the  magnetic  field,  more  or  less  ;  consequently, 
an  alternating  magnetic  field  cannot  penetrate  deeply  into  a 
solid  conductor,  but  a  kind  of  screening  effect  is  produced, 
which  makes  solid  masses  of  iron  unsuitable  for  alternating 
fields,  and  necessitates  the  use  of  laminated  iron  or  iron 
wire  as  the  carrier  of  magnetic  flux. 

Eddy  currents  are  true  electric  currents,  though  flowing 
in  minute  circuits ;  and  they  follow  all  the  laws  of  electric 
circuits. 

Their  E.M.F.  is  proportional  to  the  intensity  of  magneti- 
zation, (B,  and  to  the  frequency,  N. 

Eddy  currents  are  thus  proportional  to  the  magnetization, 
(B,  the  frequency,  Ar,  and  to  the  electric  conductivity,  y,  of 
the  iron  ;  hence,  can  be  expressed  by 

/  =  (3  y  (B  N. 

The  power  consumed  by  eddy  currents  is  proportional  to 
their  square,  and  inversely  proportional  to  the  electric  con- 
ductivity, and  can  be  expressed  by 


130  ALTERNATING-CURRENT  PHENOMENA.          [§87 

or,  since,  <B7V  is  proportional  to  the  induced  E.M.F.,  E,  in 
the  equation 

E  =  V27r,S 


it  follows  that,  The  loss  of  power  by  eddy  currents  is  propor- 
tional to  the  square  of  the  E.M.F.,  and  proportional  to  tJie 
electric  conductivity  of  the  iron  ;  or, 

W=  aEzy. 

Hence,  that  component  of  the  effective  conductance 
which  is  due  to  eddy  currents,  is 

W 

g=  —  =  ar, 

that  is,  The  equivalent  conductance  due  to  eddy  currents  in 
the  iron  is  a  constant  of  the  magnetic  circuit  ;  it  is  indepen- 
dent of  ^LM..~F.i  frequency,  etc.,  but  proportional  to  the  electric 
conductivity  of  the  iron,  y. 

87.  Eddy  currents,  like  magnetic  hysteresis,  cause  an 
advance  of  phase  of  the  current  by  an  angle  of  advance,  (3  ; 
but,  unlike  hysteresis,  eddy  currents  in  general  do  not  dis- 
tort the  current  wave. 

The  angle  of  advance  of  phase  due  to  eddy  currents  is, 

sin/2  =  £, 

y 

where  y  =  absolute,  admittance  of  the  circuit,  g  =  eddy 
current  conductance. 

While  the  equivalent  conductance,  g,  due  to  eddy  cur- 
rents, is  a  constant  of  the  circuit,  and  independent  of 
E.M.F.,  frequency,  etc.,  the  loss  of  power  by  eddy  currents 
is  proportional  to  the  square  of  the  E.M.F.  of  self-induction, 
and  therefore  proportional  to  the  square  of  the  frequency 
and  to  the  square  of  the  magnetization. 

Only  the  energy  component,  g  E,  of  eddy  currents,  is  of 
interest,  since  the  wattless  component  is  identical  with  the 
wattless  component  of  hysteresis,  discussed  in  a  preceding 
chapter. 


88,  89]        FOUCAULT  OR  EDDY  CURRENTS. 


131 


d 


88.  To  calculate  the  loss  of  power  by  eddy  currents  — 

Let  V  =  volume  of  iron  ; 

(B  =  maximum  magnetic  induction ; 
N  =•  frequency^ 

y    =  electric  conductivity  of  iron  ; 
£    =  coefficient  of  eddy  currents. 

The  loss  of  energy  per  cm3,  in  ergs  per  cycle,  is 

W  =  e  y  N  (B2  ; 
hence,  the  total  loss  of  power  by  eddy  currents  is 

W  =  e  y  V 'N'2  (B2  10  -  7  watts, 
and  the  equivalent  conductance  due  to  eddy  currents  is 

a=   W=   lOey/  _  .507  cy/ 
£2      2  7T2  Sn*          Sn2 

where : 

/  =  length  of  magnetic  circuit, 

S  =  section  of  magnetic  circuit, 

n  =  number  of  turns  of  electric  circuit. 

The  coefficient  of  eddy  currents,  c, 
depends  merely  upon  the  shape  of  the 
constituent  parts  of  the  magnetic  cir- 
cuit ;  that  is,  whether  of  iron  plates 
or  wire,  and  the  thickness  of  plates  or 
the  diameter  of  wire,  etc. 

The  two  most  important  cases  are : 

(a).    Laminated  iron. 
(b).    Iron  wire. 

89.  (a).    Laminated  Iron. 
Let,  in  Fig.  79, 

d   =  thickness  of  the  iron  plates ; 

(B  =  maximum  magnetic  induction  ; 

N  =  frequency  ;  Fig.  79. 

y    =  electric  conductivity  of  the  iron. 


132  ALTERNATING-CURRENT  PHENOMENA,          [§89 

Then,  if  x  is  the  distance  of  a  zone,  d  x,  from  the  center 
of  the  sheet,  the  conductance  of  a  zone  of  thickness,  dx, 
and  of  one  cm  length  and  width  is  ydx  ;  and  the  magnetic 
flux  cut  by  this  zone  is  (Bx.  Hence,  the  E.M.F.  induced  in 
this  zone  is 

V2  WVCBx,  in  C.G.S.  units. 


This  E.M.F.  produces  the  current : 

d  I  =  S£ydx  =  V2  TrN($>  y  x  d  x,  in  C.G.S.  units, 

provided  the  thickness  of  the  plate  is  negligible  as  compared 
with  the  length,  in  order  that  the  current  may  be  assumed 
as  flowing  parallel  to  the  sheet,  and  in  opposite  directions 
on  opposite  sides  of  the  sheet. 

The  power  consumed  by  the  induced   current   in   this 
zone,  dx,  is 

dP  =  8£df=27r*lV2W  y  xVx,  in  C.G.S.  units  or  ergs  per  second, 

and,  consequently,  the  total  power  consumed  in  one  cm2  of 
the  sheet  of  thickness,  d,  is 

,    d  ,    d 

/T  ¥  /*     3F 

d  J      d 

V  ~  2 

7T2J?V2<B2y^/8 

= — - — ,  in  C.G.S.  units  ; 

o 

the  power  consumed  per  cm3  of  iron  is,  therefore, 

i, 

- ,  in  C.G.S.  units  or  erg-seconds, 


and  the  energy  consumed  per  cycle  and  per  cm3  of  iron  is 


The  coefficient  of  eddy  currents  for  laminated  iron  is, 
therefore, 


=  ---«  1.645  d\ 
6 


§90] 


FOUCAULT  OR   EDDY  CURRENTS. 


133 


where  y  is  expressed  in  C.G.S.  units.     Hence,  if  y  is  ex- 
pressed in  practical  units  or  10 ~9  C.G.S.  units, 


c  = 


i=  1.645  d*  10- 9. 

ll;. 


Substituting  for  tire  conductivity  of  sheet  iron  the  ap- 

proximate value, 

y  =  *0» 

we  get  as  the  coefficient  of  eddy  currents  for  laminated  iron, 
c  =  lS</aiO-»  =  1.645  d*  10-  9- 


loss  of  energy  per  cm3  and  cycle, 

W=<LyNW  =  -V2y^(B210-9  =  1.645  d*yN<&  10  ~9  ergs 

=  1.645  tt^NW  10-  4  ergs; 
or,        W=  ey-AWlO-7  =  1.645  d*  N  <&  10  ~n  joules; 

loss  of  power  per  cm3  at  frequency,  N, 

p  =  NW=  eyW2(B210-7  =  1.645  </Wa(Ba.  10  ~u  watts; 
total  loss  of  power  in  volume,   F, 

P  =  Vp  =  1.645  F^2^2^2^-11  watts. 

As  an  example, 

</=  1mm  =  .1  cm;  ^=100;&  =  5000;    r=  1000  cm8. 
€  =  1,645  X  10-11; 
W=  4110  ergs 

=  .000411  joules; 
p  =  .0411  watts; 
P  =•  41.1  watts. 

90.    (b).  Iron  Wire. 

Let,  in  Fig.  80,  d  = 
diameter  of  a  piece  of 
iron  wire  ;  then  if  x  is 
the  radius  of  a  circular 
zone  of  thickness,  <^x, 
and  one  cm  in  length, 
the  conductance  of  this  Fl-g.  80. 


134  ALTERNATING-CURRENT  PHENOMENA.  [§  9O 

<\ 

zone  is,  y^/x/2  TT  x,  and  the  magnetic  flux  inclosed  by  the 
zone  is  (B  x2  w. 

Hence,  the  E.M.F.  induced  in  this  zone  is  : 


=  V2  -^N®)  x2,  in  C.G.S.  units, 
and  the  current  produced  thereby  is, 


)  in  C.G.S.  units. 


The  power  consumed  in  this  zone  is,  therefore, 

dP=  §EdI  =  ^yN^^x^doc,  in  C.G.S.  units 

consequently,  the  total  power  consumed  in  one  cm  length 
of  wire  is 


=   fa  </^=7r3y^2 

*/o  o 

2^2^4,  in  C.G.S.  units. 


Since  the  volume  of  one  cm  length  of  wire  is 


the  power  consumed  in  one  cm3  of  iron  is 

X     P  -7T2 


-2,  in  C.G.S.  units  or  erg-seconds, 
v         ID 

and  the  energy  consumed  per  cycle  and  cm3  of  iron  is 

W  =  ^-=  pyTVWergs. 
Therefore,  the  coefficient  of  eddy  currents  for  iron  wire  is 


or,  if  y  is  expressed  in  practical  units,  or  10  ~9  C.G.S.  units, 
€  =  ^^210~9  =  .617 


§91]  FO  UCA  UL  T  OR   EDD  Y  CURRENTS.  135 

Substituting  y  =  105, 

we  get  as  the  coefficient  of  eddy  currents  for  iron  wire, 

^  ' 

c  =  Z.  </a  10-9  =^617  d*  10-9. 

The    loss    of    energy  per   cm3    of    iron,   and    per    cycle 
becomes 


09  =  .617  </2 

=  .617  </  W&2  10~4  ergs, 

=  e  7  NW  10-7  =  .617  d*  NW  10-11  joules  ; 

loss  of  power  per  cm3,  at  frequency,  N, 

p=  Nh  =  ey^V2(B210-7  =  .617  ^2^2(B210-n  watts; 
total  loss  of  power  in  volume,   V, 

P=  F/  =  .617  F^2^2(B210-n  watts. 
As  an  example, 
d  =  1  mm,  =  1  cm  ;  N=  100  ;  &  =  5,000  ;    V=  1,000  cm3. 

Then, 

e  =  .617  X  10-11, 

W=  1540  ergs  =  .000154  joules, 
p  =  .0154  watts, 
P  =  15.4  watts, 

hence  very  much  less  than  in  sheet  iron  of  equal  thickness. 

91.    Comparison  of  sheet  iron  and  iron  wire. 

If 

//!  =  thickness  of  lamination  of  sheet  iron,  and 
d%  =  diameter  of  iron  wire, 

the  eddy-coefficient  of  sheet  iron  being 

€l  =  ^2io-9, 

and  the  eddy  coefficient  of  iron  wire 


136 


ALTERNATING-CURRENT  PHENOMENA. 


92 


the  loss  of  power  is  equal   in   both  —  other   things   being 
equal  —  if  Cj  =  e2  ;  that  is,  if, 


or'  </,   =  1.63  4. 

It  follows  that  the  diameter  of  iron  wire  can  be  1.63 
times,  or,  roughly,  If  as  large  as  the  thickness  of  laminated 
iron,  to  give  the  same  loss  of  energy  through  eddy  currents. 


Fig.  81. 

92.    Demagnetizing,  or  screening  effect  of  eddy  currents. 

The  formulas  derived  for  the  coefficient  of  eddy  cur- 
rents in  laminated  iron  and  in  iron  wire,  hold  only  when 
the  eddy  currents  are  small  enough  to  neglect  their  mag- 
netizing force.  Otherwise  the  phenomenon  becomes  more 
complicated;  the  magnetic  flux  in  the  interior  of  the  lam- 
ina, or  the  wire,  is  not  in  phase  with  the  flux  at  the  sur- 
face, but  lags  behind  it.  The  magnetic  flux  at  the  surface 
is  due  to  the  impressed  M.M.F.,  while  the  flux  in  the  inte- 
rior is  due  to  the  resultant  of  the  impressed  M.M.F.  and  to 
the  M.M.F.  of  eddy  currents  ;  since  the  eddy  currents  lag 
90°  behind  the  flux  producing  them,  their  resultant  with 
the  impressed  M.M.F.,  and  therefore  the  magnetism  in  the 


§92]  FOUCAULT  OR   EDDY  CURRENTS.  13T 

interior,  is  made  lagging.  Thus,  progressing  from  the  sur- 
face towards  the  interior,  the  magnetic  flux  gradually  lags 
more  and  more  in  phase,  and  at  the  same  time  decreases 
in  intensity.  While  the  complete  analytical  solution  of  this 
phenomenon  is  beyond  the  scopjj  of  this  book,  a  determina- 
tion of  the  magnitude  of  this  demagnetization,  or  screening 
effect,  sufficient  to  determine  whether  it  is  negligible,  or 
whether  the  subdivision  of  the  iron  has  to  be  increased 
to  make  it  negligible,  can  be  made  by  calculating  the  maxi- 
mum magnetizing  effect,  which  cannot  be  exceeded  by  the 
eddys. 

Assuming  the  magnetic  density  as  uniform  over  the 
whole  cross-section,  and  therefore  all  the  eddy  currents  in 
phase  with  each  other,  their  total  M.M.F.  represents  the 
maximum  possible  value,  since  by  the  phase  difference  and 
the  lesser  magnetic  density  in  the  center  the  resultant 
M.M.F.  is  reduced. 

In  laminated  iron  of  thickness  /,  the  current  in  a  zone 
of  thickness,  dx  at  distance  x  from  center  of  sheet,  is  : 


dl=  -Z-irN&jxdx  units  (C.G.S.) 

=  V2  TT  N&jxdx  10  -  8  amperes  ; 
hence  the  total  current  in  sheet  is 


f= 

o 

=  ^JL  N&jl*  10  -  8  amperes. 
8 

Hence,  the  maximum  possible  demagnetizing  ampere-turns 
acting  upon  the  center  of  the  lamina,  are 


V-* 

=  .555  7V(B  /2 10  ~3  ampere-turns  per  cm. 

Example  :     d  =  .1  cm,     N =  100,     (B  =  5,000, 
or  /  =  2.775  ampere-turns  per  cm. 


138         ALTERNATING-CURRENT  PHENOMENA.    [§§93,94 

93.    In  iron  wire  of  diameter  /,  the  current  in  a  tubular 
zone  of  dx  thickness  and  x  radius  is 

A/2 

dl=  -t^7r  N&jx  dx  10  -  8  amperes; 

2i 

hence,  the  total  current  is 

I=  C^di—vN(&j\^-^ 

Jo  2i  */  o 

10  -  8  amperes. 


Hence,  the  maximum  possible  demagnetizing  ampere-turns, 
acting  upon  the  center  of  the  wire,  are 

A/9 

/=  v^7r  JV(By/210-8  =  .27757V^(Sy/210-8 
16 

=  .2775  N®>  /2  10  -  8  ampere-turns  per  cm. 

For  example,  if  /  =  .1  cm,  N=  100,  ffi  =  5,000,  then 
f=  1,338  ampere-turns  per  cm;  that  is,  half  as  much  as  in 
a  lamina  of  the  thickness  /. 

94.  Besides  the  eddy,  or  Foucault,  currents  proper,  which 
flow  as  parasitic  circuits  in  the  interior  of  the  iron  lamina 
or  wire,  under  certain  circumstances  eddy  currents  also 
flow  in  larger  orbits  from  lamina  to  lamina  through  the 
whole  magnetic  structure.  Obviously  a  calculation  of  these 
eddy  currents  is  possible  only  in  a  particular  structure. 
They  are  mostly  surface  currents,  due  to  short  circuits 
existing  between  the  laminae  at  the  surface  of  the  magnetic 
structure. 

Furthermore,  eddy  currents  are  induced  outside  of  the 
magnetic  iron  circuit  proper,  by  the  magnetic  stray  field 
cutting  electric  conductors  in  the  neighborhood,  especially 
when  drawn  towards  them  by  iron  masses  behind,  in  elec- 
tric conductors  passing  through  the  iron  of  an  alternating 
field,  etc.  All  these  phenomena  can  be  calculated  only  in 
particular  cases,  and  are  of  less  interest,  since  they  can 
easily  be  avoided. 


§95]  FOUCAULT  OR  EDDY  CURRENTS.  139 

Eddy  Currents  in  Conductor,  and   Unequal  Current 
Distribution. 

95.  If  the  electric  conductor  has  a  considerable  size,  the 
alternating  magnetic  field,  inputting  the  conductor,  may 
set  up  differences  of  potential  between  the  different  parts 
thereof,  thus  giving  rise  to  local  or  eddy  currents  in  the 
copper.  This  phenomenon  can  obviously  be  studied  only 
with  reference  to  a  particular  case,  where  the  shape  of  the 
conductor  and  the  distribution  of  the  magnetic  field  are 
known. 

Only  in  the  case  where  the  magnetic  field  is  produced 
by  the  current  flowing  in  the  conductor  can  a  general  solu- 
tion be  given.  The  alternating  current  in  the  conductor 
produces  a  magnetic  field,  not  only  outside  of  the  conductor, 
but  inside  of  it  also  ;  and  the  lines  of  magnetic  force  which 
•close  themselves  inside  of  the  conductor  induce  E.M.Fs. 
in  their  interior  only.  Thus  the  counter  E.M.F.  of  self- 
inductance  is  largest  at  the  axis  of  the  conductor,  and  least 
at  its  surface ;  consequently,  the  current  density  at  the 
surface  will  be  larger  than  at  the  axis,  or,  in  extreme  cases, 
the  current  may  not  penetrate  at  all  to  the  center,  or  a 
reversed  current  flow  there.  Hence  it  follows  that  only  the 
exterior  part  of  the  conductor  may  be  used  for  the  conduc- 
tion of  the  current,  thereby  causing  an  increase  of  the 
ohmic  resistance  due  to  unequal  current  distribution. 

The  general  solution  of  this  problem  for  round  conduc- 
tors leads  to  complicated  equations,  and  can  be  found  in 
Maxwell. 

In  practice,  this  phenomenon  is  observed  only  with  very 
high  frequency  currents,  as  lightning  discharges  ;  in  power 
distribution  circuits  it  has  to  be  avoided  by  either  keeping 
the  frequency  sufficiently  low,  or  having  a  shape  of  con- 
ductor such  that  unequal  current  distribution  does  not 
take  place,  as  by  using  a  tubular  or  a  stranded  conductor, 
or  several  conductors  in  parallel. 


140 


ALTERNATING-CURRENT  PHENOMENA. 


[§96 


96.  It  will,  therefore,  be  sufficient  to  determine  the. 
largest  size  of  round  conductor,  or  the  highest  frequency,, 
where  this  phenomenon  is  still  negligible. 

In  the  interior  of  the  conductor,  the  current  density 
is  not  only  less  than  at  the  surface,  but  the  current  lags 
behind  the  current  at  the  surface,  due  to  the  increased 
effect  of  self -inductance.  This  lag  of  the  current  causes  the 
magnetic  fluxes  in  the  conductor  to  be  out  of  phase  with 
each  other,  making  their  resultant  less  than  their  sum,  while 
the  lesser  current  density  in  the  center  reduces  the  total 
flux  inside  of  the  conductor.  Thus,  by  assuming,  as  a  basis 
for  calculation,  a  uniform  current  density  and  no  difference 
of  phase  between  the  currents  in  the  different  layers  of  the 
conductor,  the  unequal  distribution  is  found  larger  than  it 
is  in  reality.  Hence  this  assumption  brings  us  on  the  safe 
side,  and  at  the  same  time  simplifies  the  calculation  greatly. 

Let  Fig.  82  represent  a  cross-section  of  a  conductor  of 
radius  R,  and  a  uniform  current  density, 

7 


* 


where  /  =  total  current  in  conductor. 


Fig.  82. 

The  magnetic  reluctance  of  a  tubular  zone  of  unit  length 
and  thickness  dx,  of  radius  x,  is 


dx 


§96]  FOUCAULT  OR  EDDY  CURRENTS.  141 

The  current  inclosed  by  this  zone  is  Ix  =  ixzir,  and  there- 
fore, the  M.M.F.  acting  upon  this  zone  is 

Cc         _    4:7T      ,  4  7T2  I  '  X2 

*~"io»*:    ~ir~' 

and  the  magnetic  flux  in  this^zone  is 


<R*  10 

Hence,  the  total  magnetic  flux  inside  the  conductor  is 


10       o     .  10          10 

From  this  we  get,  as  the  excess  of  counter  E.M.F.  at  the 
axis  of  the  conductor  over  that  at  the  surface  — 

=  V27r^$10-8  =  V27r^/10-9,  per  unit  length, 


and  the  reactivity,  or  specific  reactance  at  the  center  of  the 
conductor,  becomes 

k  =  -^  =  V2  7T2  NW  10  -9. 
i 

Let  p  =  resistivity,  or  specific  resistance,  of  the  material  of 
the  conductor. 
We  have  then, 

b  _ 

,  P 

and 


the  percentage  decrease  of  current  density  at  center  over 
that  at  periphery ; 

also,  r        , 

V^2  +  p2 

the  ratio  of  current  densities  at  center  and  at  periphery. 

For  example,  if,  in  copper,  p  =  1.7  x  10~6,  and  the 
percentage  decrease  of  current  density  at  center  shall  not 
exceed  5  per  cent,  that  is  — 

r  -j-  V>&2  +  p2  =  .95  -*-  1, 
we  have,  k  =  .51  X  10  ~6 ; 


142  ALTERNATING-CURRENT  PHENOMENA.  [§97 

hence        .51  X  10  ~6  =  V2  ^  NX*  10  ~9 

or  NR*  =  36.6  ; 

hence,  when  7V=       125        100        60     33.3 

£  =      .541       .605       .781     1.05  cm. 
D  =  2  £  =     1.08       1.21       1.56       2.1  cm. 

Hence,  even  at  a  frequency  of  125  cycles,  the  effect  of 
unequal  current  distribution  is  still  negligible  at  one  cm 
diameter  of  the  conductor.  Conductors  of  this  size  are, 
however,  excluded  from  use  at  this  frequency  by  the  exter- 
nal self-induction,  which  is  several  times  larger  than  the 
resistance. 

We  thus  see  that  unequal  current  distribution  is  usually 
negligible  in  practice. 

Mutual  Inductance. 

97.  When  an  alternating  magnetic  field  of  force  includes 
a  secondary  electric  conductor,  it  induces  therein  an  E.M.F. 
which  produces  a  current,  and  thereby  consumes  energy  if 
the  circuit  of  the  secondary  conductor  is  closed. 

A  particular  case  of  such  induced  secondary  currents 
are  the  eddy  or  Foucault  currents  previously  discussed. 

Another  important  case  is  the  induction  of  secondary 
E.M.Fs.  in  neighboring  circuits ;  that  is,  the  interference  of 
circuits  running  parallel  with  each  other. 

In  general,  it  is  preferable  to  consider  this  phenomenon 
of  mutual  inductance  as  not  merely  producing  an  energy 
component  and  a  wattless  component  of  E.M.F.  in  the 
primary  conductor,  but  to  consider  explicitly  both  the  sec- 
ondary and  the  primary  circuit,  as  will  be  done  in  the 
chapter  on  the  alternating-current  transformer. 

Only  in  cases  where  the  energy  transferred  into  the 
secondary  circuit  constitutes  a  small  part  of  the  total  pri- 
mary energy,  as  in  the  discussion  of  the  disturbance  caused 
by  one  circuit  upon  a  parallel  circuit,  may  the  effect  on  the 
primary  circuit  be  considered  analogously  as  in  the  chapter 
on  eddy  currents,  by  the  introduction  of  an  energy  com- 


§97]  FOUCAULT  OR  EDDY  CURRENTS.  143 

ponent,  representing  the  loss  of  power,  and  a  wattless 
component,  representing  the  decrease  of  self-inductance. 

Let-  j 

x  =  2  TT  N  L  =  reactance  'of  main  circuit;  that  is,  L  = 

HI 

total  number  of  interiinkages  with  the  main  conductor,  of 
the  lines  of  magnetic  force  produced  by  unit  current  in 
that  conductor  ; 

x^  =  27rArZj  =  reactance  of  secondary  circuit  ;  that  is, 
Ll  =  total  number  of  interlinkages  with  the  secondary 
•conductor,  of  the  lines  of  magnetic  force  produced  by  unit 
current  in  that  conductor; 

xm  =  2  TT  N  Lm  —  mutual  inductance  of  circuits  ;  that  is, 
Lm  =  total  number  of  interlinkages  with  the  secondary 
conductor,  of  the  lines  of  magnetic  force  produced  by  unit 
•current  in  the  main  conductor,  or  total  number  of  inter- 
linkages with  the  main  conductor  of  the  lines  of  magnetic 
force  produced  by  unit  current  in  the  secondary  conductor. 
Obviously  :  x 


*  As  coefficient  of  self-inductance  L,  L',  the  total  flux  surrounding  the 
conductor  is  here  meant.  Quite  frequently  in  the  discussion  of  inductive 
apparatus,  especially  of  transformers,  that  part  of  the  magnetic  flux  is  denoted 
self-inductance  of  the  one  circuit  which  surrounds  this  circuit,  but  not  the  other 
.circuit;  that  is,  which  passes  between  both  circuits.  Hence,  the  total  self- 
inductance,  Z,  is  in  this  case  equal  to  the  sum  of  the  self-inductance,  L  ', 
;and  the  mutual  inductance,  Lm. 

The  object  of  this  distinction  is  to  separate  the  wattless  part,  L',  of  the 
•total  self-inductance,  Z,  from  that  part,  Lm,  which  represents  the  transfer  of 
E.M.F.  into  the  secondary  circuit,  since  the  action  of  these  two  components  is 
•essentially  different. 

Thus,  in  alternating-current  transformers  it  is  customary  —  and  will  be 
•done  later  in  this  book  —  to  denote  as  the  self-inductance,  Z,  of  each  circuit 
^nly  that  part  of  the  magnetic  flux  produced  by  the  circuit  which  passes 
between  both  circuits,  and  thus  acts  in  "  choking  "  only,  but  not  in  transform- 
ing; while  the  flux  surrounding  both  circuits  is  called  mutual  inductance,  or 
useful  magnetic  flux. 

With  this  denotation,  in  transformers  the  mutual  inductance,  Lm,  is  usu- 
.ally  very  much  greater  than  the  self-inductances,  //,  and  Z/,  while,  if  the 
self-inductances,  L  and  Zj  ,  represent  the  total  flux,  their  product  is  larger 
than  the  square  of  the  mutual  inductance,  Lm  ;  or 


144  ALTERNATING-CURRENT  PHENOMENA.          [§  9S 

Let  rx  =  resistance  of  secondary  circuit.     Then  the  im- 
pedance of  secondary  circuit  is 


zl  =  x    ; 

E.M.F.  induced  in  the  secondary  circuit,  EI  =  /#,„/, 
where  /  =  primary  current.   Hence,  the  secondary  current  is 


and  the  E.M.F.  induced  in  the  primary  circuit  by  the  secon 
dary  current,  7X  is 


or,  expanded, 
Hence, 


=  effective  conductance  of  mutual  inductance  ; 


. 

r?  + 

b  =  ~  Xm  *l  =  effective  susceptance  of  mutual  inductance. 

r^  +  x^ 

The  susceptance  of  mutual  inductance  is  negative,  or  of 
opposite  sign  from  the  susceptance  of  self-inductance.     Or, 

Mutual  inductance  consumes  energy  and  decreases  tJie  self- 
inductance. 


Dielectric  and  Electrostatic  Phenomena. 

98.  While  magnetic  hysteresis  and  eddy  currents  can 
be  considered  as  the  energy  component  of  inductance,  con- 
densance  has  an  energy  component  also,  called  dielectric 
hysteresis.  In  an  alternating  magnetic  field,  energy  is  con- 
sumed in  hysteresis  due  to  molecular  friction,  and  similarly, 
energy  is  also  consumed  in  an  alternating  electrostatic  field 
in  the  dielectric  medium,  in  what  is  called  dielectric  hys- 
teresis. 


$99]  FOUCAULT  OR   EDDY  CURRENTS.  145 

While  the  laws  of  the  loss  of  energy  by  magnetic  hys- 
teresis are  fairly  well  understood,  and  the  magnitude  of  the 
effect  known,  the  phenomenon  of  dielectric  hysteresis  is 
.still  almost  entirely  unknown  as  concerns  its  laws  and  the 
magnitude  of  the  effect. 

It  is  quite  probable  that  the  loss  of  power  in  the  dielec- 
tric in  an  alternating  electrostatic  field  consists  of  two  dis- 
tinctly different  components,  of  which  the  one  is  directly 
proportional  to  the  frequency,  —  analogous  to  magnetic 
hysteresis,  and  thus  a  constant  loss  of  energy  per  cycle, 
independent  of  the  frequency  ;  while  the  other  component 
is  proportional  to  the  square  of  the  frequency,  —  analogous 
to  the  loss  of  power  by  eddy  currents  in  the  iron,  and  thus 
a  loss  of  energy  per  cycle  proportional  to  the  frequency. 

The  existence  of  a  loss  of  power  in  the  dielectric,  pro- 
portional to  the  square  of  the  frequency,  I  observed  some 
time  ago  in  paraffined  paper  in  a  high  electrostatic  field  and 
at  high  frequency,  by  the  electro-dynamometer  method, 
.and  other  observers  under  similar  conditions  have  found 
the  same  result. 

Arno  of  Turin  found  at  low  frequencies  and  low  field 
strength  in  a  larger  number  of  dielectrics,  a  loss  of  energy 
per  cycle  independent  of  the  frequency,  but  proportional  to 
the  1.6th  power  of  the  field  strength,  —  that  is,  following 
the  same  law  as  the  magnetic  hysteresis, 


This  loss,  probably  true  dielectric  static  hysteresis,  was 
•observed  under  conditions  such  that  a  loss  proportional  to 
the  square  of  density  and  frequency  must  be  small,  while  at 
high  densities  and  frequencies,  as  in  condensers,  the  true 
dielectric  hysteresis  may  be  entirely  obscured  by  a  viscous 
loss,  represented  by  W-&  = 


99.  If  the  loss  of  power  by  electrostatic  hysteresis  is 
proportional  to  the  square  of  the  frequency  and  of  the  field 
intensity,  —  as  it  probably  nearly  is  under  the  working  con- 


146  ALTERNATING-CURRENT  PHENOMENA.  [§99 

ditions  of  alternating-current  condensers,  —  then  it  is  pro- 
portional to  the  square  of  the  E.M.F.,  that  is,  the  effective 
conductance,  g,  due  to  dielectric  hysteresis  is  a  constant  ; 
and,  since  the  condenser  susceptance,  —  b  =  £',  is  a  constant 
also,  —  unlike  the  magnetic  inductance,  —  the  ratio  of  con- 
ductance and  susceptance,  that  is,  the  angle  of  difference 
of  phase  due  to  dielectric  hysteresis,  is  a  constant.  This  I 
found  proved  by  experiment. 

This  would  mean  that  the   dielectric   hysteretic   admit- 
tance of  a  condenser, 


=  g  —    c, 

where  g  =  hysteretic  conductance, 

//  =  hysteretic  susceptance  ; 

and  the  dielectric  hysteretic  impedance  of  a  condenser, 

Z  =  r  —jb'  =  r+jxc, 
where  :  r  =  hysteretic  resistance, 

occ  =  hysteretic  condensance  ; 


and  the  angle  of  dielectric  hysteretic  lag, 

tan  a  =  f  =  -  , 

' 


oc 


are  constants  of  the  circuit,  independent  of  E.M.F.  and  fre- 
quency. The  E.M.F.  is  obviously  inversely  proportional  to 
the  frequency. 

The  true  static  dielectric  hysteresis,  observed  by  Arno 
as  proportional  to  the  1.6th  power  of  the  density,  will  enter 
the  admittance  and  the  impedance  as  a  term  variable  and 
dependent  upon  E.M.F.  and  frequency,  in  the  same  manner 
as  discussed  in  the  chapter  on  magnetic  hysteresis. 

To  the  magnetic  hysteresis  corresponds,  in  the  electro- 
static field,  the  static  component  of  dielectric  hysteresis,, 
following,  probably,  the  same  law  of  1.6th  power. 

To  the  eddy  currents  in  the  iron  corresponds,  in  the 
electrostatic  field,  the  viscous  component  of  dielectric  hys- 
teresis, following  the  square  law. 


§  1OO]  FOUCAULT  OR  EDDY  CURRENTS.  14T 

To  the  phenomenon  of  mutual  inductance  corresponds, 
in  the  electrostatic  field,  the  electrostatic  induction,  or  in- 
fluence. • 

t 

100.  The  alternating  electrostatic  field  of  force  of  an 
electric  circuit  induces,  in  conductors  within  the  field  of 
force,  electrostatic  charges  by  what  is  called  electrostatic 
influence.  These  charges  are  proportional  to  the  field 
strength  ;  that  is,  to  the  E.M.F.  in  the  main  circuit. 

If  a  flow  of  current  is  produced  by  the  induced  charges, 
energy  is  consumed  proportional  to  the  square  of  the  charge ; 
that  is,  to  the  square  of  the  E.M.F. 

These  induced  charges,  reacting  upon  the  main  conduc- 
tor, influence  therein  charges  of  equal  but  opposite  phase, 
and  hence  lagging  behind  the  main  E.M.F.  by  the  angle 
of  lag  between  induced  charge  and  inducing  field.  They 
require  the  expenditure  of  a  charging  current  in  the  main 
conductor  in  quadrature  with  the  induced  charge  thereon  ; 
that  is,  nearly  in  quadrature  with  the  E.M.F.,  and  hence 
consisting  of  an  energy  component  in  phase  with  the 
E.M.F.  —  representing  the  power  consumed  by  electrostatic 
influence  —  and  a  wattless  component,  which  increases  the 
capacity  of  the  conductor,  or,  in  other  words,  reduces  its 
capacity  susceptance,  or  condensance. 

Thus,  the  electrostatic  influence  introduces  an  effective 
conductance,  g,  and  an  effective  susceptance,  b, — of  oppo- 
site sign  with  condenser  susceptance,  —  into  the  equations 
of  the  electric  circuit. 

While  theoretically  g  and  b  should  be  constants  of  the 
circuit,  frequently  they  are  very  far  from  such,  due  to 
disruptive  phenomena  beginning  to  appear  at  these  high 
densities. 

Even  the  capacity  condensance  changes  at  very  high 
potentials  ;  escape  of  electricity  into  the  air  and  over  the 
surfaces  of  the  supporting  insulators  by  brush  discharge  or 
electrostatic  glow  takes  place.  As  far  as  this  electrostatic 


148  ALTERNATING-CURRENT  PHENOMENA,          [§  101 

corona  reaches,  the  space  is  in  electric  connection  with  the 
conductor,  and  thus  the  capacity  of  the  circuit  is  deter- 
mined, not  by  the  surface  of  the  metallic  conductor,  but 
by  the  exterior  surface  of  the  electrostatic  glow  surround- 
ing the  conductor.  This  means  that  with  increasing  po- 
tential, the  capacity  increases  as  soon  as  the  electrostatic 
corona  appears  ;  hence,  the  condensance  decreases,  and  at 
the  same  time  an  energy  component  appears,  representing 
the  loss  of  power  in  the  corona. 

This  phenomenon  thus  shows  some  analogy  with  the  de- 
crease of  magnetic  inductance  due  to  saturation. 

At  moderate  potentials,  the  condensance  due  to  capacity 
can  be  considered  as  a  constant,  consisting  of  a  wattless 
component,  the  condensance  proper,  and  an  energy  com- 
ponent, the  dielectric  hysteresis. 

The  condensance  of  a  polarization  cell,  however,  begins 
to  decrease  at  very  low  potentials,  as  soon  as  the  counter 
E.M.F.  of  chemical  dissociation  is  approached. 

The  condensance  of  a  synchronizing  alternator  is  of  the 
nature  of  a  variable  quantity  ;  that  is,  the  synchronous 
reactance  changes  gradually,  according  to  the  relation  of 
impressed  and  of  counter  E.M.F.,  from  inductance  over 
zero  to  condensance. 

Besides  the  phenomena  discussed  in  the  foregoing  as 
terms  of  the  energy  components  and  the  wattless  compo- 
nents of  current  and  of  E.M.F.,  the  electric  leakage  is 
to  be  considered  as  a  further  energy  component ;  that  is, 
the  direct  escape  of  current  from  conductor  to  return  con- 
ductor through  the  surrounding  medium,  due  to  imperfect 
insulating  qualities.  This  leakage  current  represents  an 
effective  conductance,  g,  theoretically  independent  of  the 
E.M.F.,  but  in  reality  frequently  increasing  greatly  with  the 
E.M.F.,  owing  to  the  decrease  of  the  insulating  strength  of 
the  medium  upon  approaching  the  limits  of  its  disruptive 
strength. 


~§1O1]  FOUCAULT  OR  EDDY  CURRENTS.  149 

101.  In  the  foregoing,  the  phenomena  causing  loss  of 
-energy  in  an  alternating-current  circuit  have  been  dis- 
cussed ;  and  it  has  been  shown  that  the  mutual  relation 
between  current  and  E.M.F.  can  be  expressed  by  two  of 
the  four  constants  :  «•• 

Energy   component  of   E.M.F.,  in  phase  with  current,  and  = 

current  X  effective  resistance,  or  r ; 
wattless  component  of  E.M.F.,  in  quadrature  with  current,  and  = 

current  X  effective  reactance,  or  x  ; 
^energy   component   of   current,  in  phase  with   E.M.F.,  and  = 

E.M.F.  X  effective  conductance,  or^; 
wattless  component  of  current,  in  quadrature  with  E.M.F.,  and  = 

E.M.F.  X  effective  susceptance,  or  b. 

In  many  cases  the  exact  calculation  of  the  quantities, 
r,  x,  g,  b,  is  not  possible  in  the  present  state  of  the  art. 

In  general,  r,  x,  g,  b,  are  not  constants  of  the  circuit,  but 
depend  —  besides  upon  the  frequency  —  more  or  less  upon 
E.M.F.,  current,  etc.  Thus,  in  each  particular  case  it  be- 
comes necessary  to  discuss  the  variation  of  r,  x,  g,  b,  or  to 
determine  whether,  and  through  what  range,  they  can  be 
assumed  as  constant. 

In  what  follows,  the  quantities  r,  x,  g,  b,  will  always  be 
considered  as  the  coefficients  of  the  energy  and  wattless 
components  of  current  and  E.M.F.,  —  that  is,  as  the  effec- 
tive quantities,  —  so  that  the  results  are  directly  applicable 
to  the  general  electric  circuit  containing  iron  and  dielectric 
losses. 

Introducing  now,  in  Chapters  VII.  to  IX.,  instead  of 
"ohmic  resistance,"  the  term  "effective  resistance,"  etc., 
as  discussed  in  the  preceding  chapter,  the  results  apply 
also  —  within  the  range  discussed  in  the  preceding  chapter 
—  to  circuits  containing  iron  and  other  materials  producing 
energy  losses  outside  of  the  electric  conductor. 


150  ALTERNATING-CURRENT  PHENOMENA.        [§102 


CHAPTER    XII. 

DISTRIBUTED  CAPACITY,   INDUCTANCE,   RESISTANCE,  AND 

LEAKAGE. 

102.  As  far  as  capacity  has  been  considered  in  the 
foregoing  chapters,  the  assumption  has  been  made  that  the 
condenser  or  other  source  of  negative  reactance  is  shunted 
across  the  circuit  at  a  definite  point.  In  many  cases,  how- 
ever, the  capacity  is  distributed  over  the  whole  length  of  the 
conductor,  so  that  the  circuit  can  be  considered  as  shunted 
by  an  infinite  number  of  infinitely  small  condensers  infi. 
nitely  near  together,  as  diagrammatically  shown  in  Fig.  83. 

f.mimillllJlllllllliJi 
ITTTTTTTmTTTTTTTrTTTTTi 

Fig.  83.    Distributed  Capacity. 

In  this  case  the  intensity  as  well  as  phase  of  the  current,, 
and  consequently  of  the  counter  E.M.F.  of  ^inductance  and 
resistance,  vary  from  point  to  point ;  and  it  is  no  longer 
possible  to  treat  the  circuit  in  the  usual  manner  by  the 
vector  diagram. 

This  phenomenon  is  especially  noticeable  in  long-distance 
lines,  in  underground  cables,  especially  concentric  cables,  and 
to  a  certain  degree  in  the  high-potential  coils  of  alternating- 
current  transformers.  It  has  the  effect  that  not  only  the 
E.M.Fs.,  but  also  the  currents,  at  the  beginning,  end,  and 
different  points  of  the  conductor,  are  different  in  intensity 
and  in  phase. 

Where  the  capacity  effect  of  the  line  is  small,  it  may 
with  sufficient  approximation  be  represented  by  one  con- 


§  1O3]  DISTRIBUTED    CAPACITY.  151 

denser  of  the  same  capacity  as  the  line,  shunted  across  the 
line.  Frequently  it  makes  no  difference  either,  whether 
this  condenser  is  considered  as  connected  across  the  line  at 
the  generator  end,  or  at  the  'receiver  end,  or  at  the  middle. 

• 

The  best  approximation  'is  to  consider  the  line  as 
shunted  at  the  generator  and  at  the  motor  end,  by  two 
condensers  of  \  the  line  capacity  each,  and  in  the  middle 
by  a  condenser  of  \  the  line  capacity.  This  approximation, 
based  on  Simpson's  rule,  assumes  the  variation  of  the  elec- 
tric quantities  in  the  line  as  parabolic. 

If,  however,  the  capacity  of  the  line  is  considerable,  and 
the  condenser  current  is  of  the  same  magnitude  as  the 
main  current,  such  an  approximation  is  not  permissible,  but 
each  line  element  has  to  be  considered  as  an  infinitely 
small  condenser,  and  the  differential  equations  based  thereon 
integrated. 

103.  It  is  thus  desirable  to  first  investigate  the  limits 
of  applicability  of  the  approximate  representation  of  the  line 
by  one  or  by  three  condensers. 

Assuming,  for  instance,  that  the  line  conductors  are  of 
1  cm  diameter,  and  at  a  distance  from  each  other  of  50  cm, 
and  that  the  length  of  transmission  is  50  km,  we  get  the 
capacity  of  the  transmission  line  from  the  formula  — 

l.llXlO-6e/     .      ,       , 
c  = microfarads, 

4  log  nat  — — 
where 

K  =  dielectric  constant  of  the  surrounding  medium  =  1  in  air ;; 

/  =  length  of  conductor  =  5  X  106  cm.  ; 

d  =  distance  of  conductors  from  each  other  =  50  cm. ; 

B  =  diameter  of  conductor  =  1  cm. 

Since 

C  =  .3  microfarads, 

the  capacity  reactance  is 

106 


152  ALTERNATING-CURRENT  PHENOMENA.         [§  1O4 

where  N  —  frequency ;  hence,  at  N  —  60  cycles, 
x  =  8,900  ohms  ; 

and  the  charging  current  of  the  line,  at  E  =  20,000  volts, 

becomes,  g 

i0  =  —  =  2.25  amperes. 
x 

The  resistance  of  100  km  of  line  of  1  cm  diameter  is  22 
ohms  ;  therefore,  at  10  per  cent  =  2,000  volts  loss  in  the 
line,  the  main  current  transmitted  over  the  line  is 

2,000 
7  =  -—  =  91  amperes, 

representing  about  1,800  kw. 

In  this  case,  the  condenser  current  thus  amounts  to  less 
than  2^  per  cent.,  and  hence  can  still  be  represented  by  the 
approximation  of  one  condenser  shunted  across  the  line. 

If,  however,  the  length  of  transmission  is  150  km  and 
the  voltage  30,000, 

capacity  reactance  at  60  cycles,       x  =  2,970  ohms ; 

charging  current,  i0  =  10.1  amperes  ; 

line  resistance,  r  =  66  ohms  ; 

main  current  at  10  p>er  cent  loss,    /  =  45.5  amperes. 
The  condenser  current   is  thus  about  22  per  cent,  of  the 
main  current. 

At  300  km  length  of  transmission  it  will,  at  10  per  cent, 
loss  and  with  the  same  size  of  conductor,  rise  to  nearly  90 
per  cent,  of  the  main  current,  thus  making  a  more  explicit 
investigation  of  the  phenomena  in  the  line  necessary. 

In  most  cases  of  practical  engineering,  however,  the  ca- 
pacity effect  is  small  enough  to  be  represented  by  the  approx- 
imation of  one  ;  viz.,  three  condensers  shunted  across  the  line. 

104.  A.)  Line  capacity  represented  by  one  condenser 
.shunted  across  middle  of  line. 

Let  — 

Y  =  g  -\-  jb  =  admittance  of  receiving  circuit; 
z  =  r  —  j x  =  impedance  of  line ; 
bc  =  condenser  susceptance  of  line. 


§  1O5]  DISTRIBUTED   CAPACITY,  15& 

Denoting,  in  Fig.  84, 

the  E.M.F.,  viz.,  current  in  receiving  circuit  by  E,  7, 

the  E.M.F.  at  middle  of  line  by  E', 

the  E.M.F.,  viz.,  current  at  generator  by  E0,J0; 


if 


Fig.  84.    Capacity  Shunted  across  Middle  of  Line. 

We  have, 


f0    =  I-jbcEc' 


\l-\    (r->^) 
\ 


2 
jbe(r-jx) 

~~ 


I 


or,  expanding, 

/„  =^{[^+^(r* 

£„  =  £     1  +  (r  -/*)  (*•+>*)  -(r-  jx) 


=  ^  1  1  +  (r  -jx)    g  +  jt  -i-l^  (r  -jx 


105.    j5.)  Z/«^  capacity  represented  by  three  condensers, 
in  the  middle  and  at  the  ends  of  the  line. 
Denoting,  in  Fig.  85, 

the  E.M.F.  and  current  in  receiving  circuit  by  E,  7, 
the  E.M.F.  at  middle  of  line  by  Er, 


154  ALTERNATING-CURRENT  PHENOMENA,        [§  1O5 

the  current  on  receiving  side  of  line  by  7', 
the  current  on  generator  side  of  line  by  /", 
the  E.M.F.,  viz.,  current  at  generator  by  £0,  S0, 


I 


Fig.  85.    Distributed  Capacity. 

otherwise  retaining  the  same  denotations  as  in  A.), 
We  have, 


r  =  i-* 


-jx)    ff  +  jb  --          (r  _ 


7.  - 


As  will  be  seen,  the  first  terms  in  the  expression   of  E0 
and  of  I0  are  the  same  in  A.)  and  in  B.). 


§  1O6]  DISTRIBUTED    CAPACITY.  155 

106.  C.)  Complete  investigation  of  distributed  capacity, 
inductance,  leakage,  and  resistance. 

In  some  cases,  especially  in  very  long  circuits,  as  in 
lines  conveying  alternating  power  currents  at  high  potential 
over  extremely  long  distances  b^  overhead  conductors  or  un- 
derground cables,  or  with  very  feeble  currents  at  extremely 
high  frequency,  such  as  telephone  currents,  the  consideration 
of  the  line  resistance  —  which  consumes  E.M.Fs.  in  phase 
with  the  current  —  and  of  the  line  reactance  —  which  con- 
sumes E.M.Fs.  in  quadrature  with  the  current  —  is  not 
sufficient  for  the  explanation  of  the  phenomena  taking  place 
in  the  line,  but  several  other  factors  have  to  be  taken  into 
account. 

In  long  lines,  especially  at  high  potentials,  the  electro- 
static capacity  of  the  line  is  sufficient  to  consume  noticeable 
•currents.  The  charging  current  of  the  line  condenser  is 
proportional  to  the  difference  of  potential,  and  is  one-fourth 
period  ahead  of  the  E.M.F.  Hence,  it  will  either  increase 
or  decrease  the  main  current,  according  to  the  relative  phase 
•of  the  main  current  and  the  E.M.F. 

As  a  consequence,  the  current  will  change  in  intensity 
;as  well  as  in  phase,  in  the  line  from  point  to  point ;  and  the 
E.M.Fs.  consumed  by  the  resistance  and  inductance  will 
therefore  also  change  in  phase  and  intensity  from  point 
to  point,  being  dependent  upon  the  current. 

Since  no  insulator  has  an  infinite  resistance,  and  as  at 
high  potentials  not  only  leakage,  but  even  direct  escape  of 
electricity  into  the  air,  takes  place  by  "  silent  discharge,"  we 
have  to  recognize  the  existence  of  a  current  approximately 
proportional  and  in  phase  with  the  E.M.F.  of  the  line. 
This  current  represents  consumption  of  energy,  and  is 
therefore  analogous  to  the  E.M.F.  consumed  by  resistance, 
while  the  condenser  current  and  the  E.M.F.  of  inductance 
are  wattless. 

Furthermore,  the  alternate  current  passing  over  the  line 
induces  in  all  neighboring  conductors  secondary  currents, 


156  ALTERNATING-CURRENT  PHENOMENA.       [§106 

which  react  upon  the  primary  current,  and  thereby  intro- 
duce E.M.Fs.  of  mutual  inductance  into  the  primary  circuit. 
Mutual  inductance  is  neither  in  phase  nor  in  quadrature 
with  the  current,  and  can  therefore  be  resolved  into  an 
energy  component  of  mutual  inductance  in  phase  with  the 
current,  which  acts  as  an  increase  of  resistance,  and  into 
a  wattless  component  in  quadrature  with  the  current,  which 
decreases  the  self-inductance. 

This  mutual  inductance  is  by  no  means  negligible,  as, 
for  instance,  its  disturbing  influence  in  telephone  circuits 
shows. 

The  alternating  potential  of  the  line  induces,  by  electro- 
static influence,  electric  charges  in  neighboring  conductors 
outside  of  the  circuit,  which  retain  corresponding  opposite 
charges  on  the  line  wires.  This  electrostatic  influence  re- 
quires the  expenditure  of  a  current  proportional  to  the 
E.M.F.,  and  consisting  of  an  energy  component,  in  phase 
with  the  E.M.F.,  and  a  wattless  component,  in  quadrature 
thereto. 

The  alternating  electromagnetic  field  of  force  set  up  by 
the  line  current  produces  in  some  materials  a  loss  of  energy 
by  magnetic  hysteresis,  or  an  expenditure  of  E.M.F.  in 
phase  with  the  current,  which  acts  as  an  increase  of  re- 
sistance. This  electromagnetic  hysteretic  loss  may  take 
place  in  the  conductor  proper  if  iron  wires  are  used,  and 
will  then  be  very  serious  at  high  frequencies,  such  as  those 
of  telephone  currents. 

The  effect  of  eddy  currents  has  already  been  referred 
to  under  "mutual  inductance,"  of  which  it  is  an  energy 
component. 

The  alternating  electrostatic  field  of  force  expends 
energy  in  dielectrics  by  what  is  called  dielectric  hysteresis. 
In  concentric  cables,  where  the  electrostatic  gradient  in  the 
dielectric  is  comparatively  large,  the  dielectric  hysteresis 
may  at  high  potentials  consume  far  greater  amounts  of 
energy  than  the  resistance  does.  The  dielectric  hysteresis 


§  1O7]  DISTRIBUTED   CAPACITY.  157 

appears  in  the  circuit  .as  consumption  of  a  current,  whose 
component  in  phase  with  the  E.M.F.  is  the  dielectric  energy 
current,  which  may  be  considered  as  the  power  component 
of  the  capacity  current. 

Besides  this,  there-  is  the  increase  of  ohmic  resistance 
due  to  unequal  distribution  of  current,  which,  however,  is 
usually  not  large  enough  to  be  noticeable. 

107.  This  gives,  as  the  most  general  case,  and  per  unit 
length  of  line  : 

E.M.Fs.  consumed  in  phase  with  the  current  I,  and  =  rl, 
representing  consumption  of  energy,  and  due  to  : 
Resistance,  and  its  increase  by  unequal  current  distri- 
tribution ;    to   the   energy   component   of   mutual 
inductance ;    to  induced  currents ;    to  the  energy 
component   of    s  elf -induct am  e  ;    or   to   electromag- 
netic hysteresis. 
E.M.Fs.  consumed  in  quadrature  with  the  current  I,  and 

==  x  I,  wattless,  and  due  to  : 
Self-inductance,  and  Mutual  inductance. 
Currents  consumed  in  pJiase  with   the  E.M.F.,   E,    and 
=  gE,  representing  consumption  of  energy,  and 
due  to  : 

Leakage  through   the   insulating  material,   including 
silent    discharge ;     energy    component    of    electro- 
static influence  ;  energy  component  of  capacity,  or 
of  dielectric  hysteresis. 
Currents  consumed  in  quadrature  to  the  E.M.F.,  E,  and 

=  bE,  beingx  wattless,  and  due  to  : 
Capacity  and  Electrostatic  influence. 

Hence  we  get  four  constants  :  — 

Effective  resistance,  r, 
Effective  reactance,  x, 
Effective  conductance,  g, 
Effective  susceptance,  b  =  —  br> 


158     ALTERNATING-CURRENT  PHENOMENA.    [§§  1O8,  1O9 

per  unit  length  of  line,  which  represent  the  coefficients,  per 
unit  length  of  line,  of 

E.M.F.  consumed  in  phase  with  current  ; 
E.M.F.  consumed  in  quadrature  with  current  ; 
Current  consumed  in  phase  with  E.M.F.  ; 
Current  consumed  in  quadrature  with  E.M.F. 

108.  This  line  we  may  assume  now  as  feeding  into  a 
receiver  circuit  of  any  description,  and  determine  the  current 
and  E.M.F.  at  any  point  of  the  circuit. 

That  is,  an  E.M.F.  and  current  (differing  in  phase  by  any 
desired  angle)  may  be  given  at  the  terminals  of  receiving  cir- 
cuit. To  be  determined  are  the  E.M.F.  and  current  at  any 
point  of  the  line  ;  for  instance,  at  the  generator  terminals. 

Or,  ^i  =  r^  —jx±\ 

the  impedance  of  receiver  circuit,  or  admittance, 

^1=^1  +>£i, 

and  E.M.F.,  E0,  at  generator  terminals  are  given.     Current 
and  E.M.F.  at  any  point  of  circuit  to  be  determined,  etc. 

109.  Counting  now  the  distance,  x,  from  a  point,  0,  of 
the  line  which  has  the  E.M.F., 


fit  and  the  current  :  I\  =  i\ 

and  counting  x  positive  in  the  direction  of  rising  energy, 
and  negative  in  the  direction  of  decreasing  energy,  we  have 
at  any  point,  X,  in  the  line  differential,  </x  : 

Leakage  current  :  Egdx-, 
Capacity  current  :    —  /  E  bc  d  x  ; 

hence,  the  total  current  consumed  by  the  line  element,  </x, 


s  -,  or, 


(1) 


E.M.F.  consumed  by  resistance, 

E.M.F.  consumed  by  reactance,    —jlxd*\ 


§  110]  DISTRIBUTED   CAPACITY.  159 

hence,  the  total  E.M.F.  consumed  in  the  line  element,  </x,  is 

dE    =  I(r  —  jx}  </x,   or, 

dE    =I, 

d* 

A 
These  fundamental  differential  eqtiations : 

(I  •*•  77*     /  •  \  /-<  \ 

dE         ..         .... 


are  symmetrical  with  respect  to  /  and  E. 
Differentiating  these  equations : 


rfx 

^  '  <3> 


and  substituting  (3)  in  (1)  and  (2),  we  get  : 

(4) 

(5) 


the  differential  equations  of  E  and  I. 

110.  These  differential  equations  are  identical,  and  con- 
sequently I  and  E  are  functions  differing  by  their  limiting 
conditions  only. 

These  equations,  (4)  and  (5),  are  of  the  form  : 


(6) 

and  are  integrated  by 

w  =  #  erx, 

where  e  is  the  basis  of    natural  logarithms  ;  for,  differen- 
tiating this,  we  get, 


160  A  L  TERN  A  TING-CURRENT  PHENOMENA .      [  §  1 1 0 

hence,  v*  =  (g  -  j  bc)  (r  -  J  x)  ; 


or,  v  =  i  V  (g  -  jbc)  (r  -  j  x)  ; 

hence,  the  general  integral  is  : 

»-»«+«+ J.--T  (8) 

where  a  and  b  are  the  two  constants  of  integration ;  sub- 
stituting 

*  =  a-y/3  (9) 

into  (7),  we  have, 


or, 

c?-p=gr-xbe; 
2aft=gx  +  bcr-, 
therefore, 

(10) 


=  Vl/2 

/  —  -  —  —======  -    f     v11/ 

=   Vl/2 


substituting  (9)  into  (8)  : 

w  =  af.(°-~J^*  -f  ^€ 


+y  sin/3x)  ; 
w  =  (^eax  +  ^€~ax)  cos/?x  —  >  (rtcax  —  ^c-ax)  sin  /?x  (12) 

which  is  the  general  solution  of  differential  equations  (4) 
and  (5) 

Differentiating  (8)  gives  : 


hence,  substituting,  (9)  : 

=  (a  —  //?)  {(#cax  —  £e~ax)  (cos  /?x  —  / 


Substituting  now  /  for  w,  and  substituting  (13)  in  (1), 
and  writing, 


§§  111,  112]         DISTRIBUTED   CAPACITY 

we  get  : 


161 


(l-       *      {(At"+l 

> 

* 

*~/0 

sin  /?x}; 
1 

•sin    R*\- 

(14) 


where  ^4  and  B  are  the  constants  of  integration. 

These  are  the  general  integral  equations  of  the  problem. 


111.  If  — 

7t  =  /!  -f  y //  is  the  current 


.  = 


by  substituting  (15)  in  (14),  we  get : 
2A=  {(a  ,\  +  ft  //)  +  (^  +  bc  ^' 

H-  /  {(«  ''/  -  ^  *i)  +  (^  V  - 
2  ^  =   {(a  4  +  /8  //) 


(16) 


a  and  ft  being  determined  by  equations  (11). 


112.    If  Z  =  J?  —  j X  is  the  impedance  of  the  receiver 
circuit,  E0  —  e0  -f  j '  ej  is  the  E.M.F.  at  dynamo  terminals 
(17),  and  /  =  length  of  line,  we  get 
at  x  =  0, 


K  _ 

— 


hence 

or 

At  x  =  /, 


A-  B 

-  ^T 
g  -  Jbc 

E 


A  —  B      a  —  j  $ 
A  +  £     g-jbc  ' 


E.= 


1 


g  — 


—  {(At*1  -  &t-**ivQ*pt—J(A**l+-JB'j 


sin/3/}. 


(18) 


(19) 


162  ALTERNATING-CURRENT  PHENOMENA.       [§113 

Equations  (18)  and  (19)  determine  the  constants^  and  B, 
which,  substituted  .in  (14),  give  the  final  integral  equations. 

The  length,  X0  =  %-*/  ft  is  a  complete  wave  length  (20), 
which  means,  that  in  the  distance  27T//3  the  phases  of  cur- 
rent and  E.M.F.  repeat,  and  that  in  half  this  distance,  they 
are  just  opposite. 

Hence  the  remarkable  condition  exists  that,  in  a  very 
long  line,  at  different  points  the  currents  at  the  same  time 
flow  in  opposite  directions,  and  the  E.M.Fs.  are  opposite. 

113.  The  difference  of  phase,  £,  between  current,  /,  and 
E.M.F.,  E,  at  any  point,  x,  of  the  line,  is  determined  by 
the  equation, 

D  (cos  o>  -f-  /  sin  u>)  =  — , 

where  D  is  a  constant. 

Hence,  £  varies  from  point  to  point,  oscillating  around  a 
medium  position,  wx,  which  it  approaches  at  infinity. 

This  difference  of  phase,  £><*,  towards  which  current  and 
E.M.F.  tend  at  infinity,  is  determined  by  the  expression, 

D  (cos  w*  +  /sin  £*)  =   /— 

// 

or,  substituting  for  E  and  /their  values,  and  since  e~a*  =  0, 
and  A  cax  (cos  ft  x  —  /  sin  ft  x),  a  cancels,  and 

£>(cosZ>*  +  / sin  woe)  =  a~//? 


g  -  jc 
=  (ag  +  fibe)  -/(a  be  -  ftg)  . 


hence,  tanS,  =  -  af     ,*\  (21) 

«^H-  P&c 

This  angle,  o>x,  =  0  ;  that  is,  current  and  E.M.F.  come 
more  and  more  in  phase  with  each  other,  when 
^c  ~  $g=  0;  that  is, 
a  -r-  ft  =  g  -^  bc  ,  or, 


§114]  DISTRIBUTED    CAPACITY.  163 

substituting  (10),  gives, 


hence,  expanding,         r  -4-  x  —  g  -f-  bc  ;  (22) 

•*• 

that  is,  the  ratio  of  resistance  to  inductance  equals  the  ratio 
of  leakage  to  capacity. 

This  angle,  woe,  =  45°  ;  that  is,  current  and  E.M.F.  differ 
by  Jth  period,  if  —  a  bc  +  $g  =  <*.£  +  ft  bc  ;  or, 

a   =  ^  +  g  . 

(3       bc+g' 
which  gives  :  rg+  xbc  =  0.  (23) 

That  is,  two  of  the  four  line  constants  must  be  zero  ;  either 
g  and  x,  or  g  and  bc  . 

114.  As  an  instance,  in  Fig.  86  a  line  diagram  is  shown, 
with  the  distances  from  the  receiver  end  as  abscissae. 
The  diagram  represents  one  and  one-half  complete  waves, 
and  gives  total  effective  current,  total  E.M.F.,  and  differ- 
ence of  phase  between  both  as  function  of  the  distance  from 
receiver  circuit  ;  under  the  conditions, 

E.M.F.  at  receiving  end,  10,000  volts  ;  hence,  E^  =  el  =  10,000; 

current  at  receiving  end,  65  amperes,  with  an  energy  co- 

efficient of  .385, 
that  is,  1=  h+jij  =  25  +  60  y; 

line  constants  per  unit  length, 

g  =  2  x  10-5, 
bc  =  20  x  10-5; 


length  of  line  corresponding  to 
x0  =  L  =  —  -  =  221.5  =•      one  complete  period  of  the  wave 

*  [     of  propagation. 

A  =  1.012  —  1.206  y, 
B  =    .812  +    .794  / 


164 


AL  TERNA  TING-CURRENT  PHENOMENA.        [§115 


These  values,  substituted,  give, 

7  =     {€«x  (47.3  cos  /3x  -f  27.4  sin  fix)  —  e-ax 

(22.3  cos  fix  +  32.6  sin  fix)} 
+  j  {eax  (27.4  cos  fix  —  47.3  sin  /?x)  -f  e-ax 

(32.6  cos  /3x  —  22.3  sin  /3x)}; 
E  =     {cax  (6450  cos  /3x  +  4410  sin  /?x)  +  c-a) 

(3530  cos  /3x  —  4410  sin  /3x)} 
_j_y|cax  (4410  cos  fix  —  6450  sin  J3x)  —  c-a; 
(4410  cos  fix  +  3530  sin  0x)}; 


tan  oi,  = 


=  -  .073, 


=  -  4.2°. 


—  ^ 

+  30 

*EsNeCf 

\ 
\ 

1 

OLT? 

0,000 

*20 

i 

\ 

..J. 

+  10' 

/ 

i 

^ 

' 

"\ 

L 

0 

(i 

\h 

/* 

\ 

X 

'e 

^*"" 

^~- 

3 

»7000 

—10 

\ 

s-.- 

/ 

2,000 

-20 

i 

\ 

/' 

/ 

_!« 

23,000 

—30 

/ 

^** 

' 

^ 

^  "   • 

—  40  ^ 

i 

f 

MW 

260    2 

f.'ooo 

4000 
2.000 

1 

-*? 

/ 

240    3 

I 

^ 

/' 

*?0    2 

/ 

/ 

200    20,000 

x  — 

*>. 

/ 

9* 

190    18.000 

/ 

*^ 

^^ 

_^  . 

/ 

•  80     l.«,000 

/ 

1 

s 

,,0 

4  000 

/ 

1  —  ~ 

V 

/ 

7 

120 

,.000 

/ 

\ 

^ 

/ 

too 

0,000 

X 

y 

x^ 

^ 

r  =  l 

s  =4 

80 

8.000 

/ 

[=5 

o.oo 

It 

60 

6,000 

\ 

/ 

t! 

.000 

40 

4,000 

\ 

20 

2,000 

o 

^ 

i 

3L 

T 

5L 
4~ 

3L 

T 

Fig.  86. 


-§115]  DISTRIBUTED    CAPACITY.  165 

115.    The  following  are  some  particular  cases  : 
A.)    Open  circuit  at  end  of  lines  : 
x  =  0  :"  /!  =  0. 


hence, 

E= 


x  _  £-ax)cos£x  _y 
*r-*fft 

'.)    Line  grounded  at  end: 

x  =  0 :  E,  =  0. 


'X  _   £-«x)cos£x   _y(£ax  +   €-ax)  s'm  ft  x 

IX  +  e~ax)  cos/3x  — y(cax  —  c-ax)sin/?x}. 


C.)    Infinitely  long  conductor  : 

Replacing  x  by  —  x,  that  is,  counting  the  distance  posi- 
tive in  the  direction  of  decreasing  energy,  we  have, 

x  =  oo  :  1=  0,  E  =  0; 
hence 

B  =  0, 
and         E  =  —  1_  A  £_ax  (CQS  ^  x        gin 

.r—  y^ 

/  =  —  i—  ^e-ax(cos/3x  +ysin/3x), 


involving  decay  of  the  electric  wave. 

The  total  impedance  of  the  infinitely  long  conductor  is 


8- 
__  (a  -j 


166  ALTERNATING-CURRENT  PHENOMENA.       [§115 

The  infinitely  long  conductor  acts  like  an  impedance 

that  is,  like  a  resistance 

J\.  =  — , 

combined  with  a  reactance 


g*  +  V 

We  thus  get  the  difference  of  phase  between   E.M.F.. 
and  current, 


which  is  constant  at  all  points  of  the  line. 
If  g  =  0,  x  =  0,  we  have, 


hence, 

tan  oi  =  l,  or, 

o>  =  45°  ; 
that  is,  current  and  E.M.F.  differ.  by  Jth  period. 

D.)    Generator  feeding  into  closed  circuit  : 
Let  x  =  0  be  the  center  of  cable  ;  then, 

E^  =  —  E_*  ;          hence  :    E  =  0  at  x  =  0  ; 

/x   «=/-*; 

which  equations  are  the  same  as  in  By  where  the  line  is 
grounded  at  x  =  0. 


§116,117]     ALTERNATING-CURRENT  TRANSFORMER.     16T 


CHAPTER    XIII. 

THE    ALTERNATING-CURRENT    TRANSFORMER. 

116.  The  simplest  alternating-current  apparatus  is  the 
transformer.     It  consists  of  a  magnetic  circuit  interlinked 
with  two  electric  circuits,  a  primary  and  a  secondary.     The 
primary  circuit  is  excited  by  an  impressed  E.M.F.,  while  in 
the  secondary  circuit  an  E.M.F.  is  induced.     Thus,  in  the 
primary  circuit  power  is  consumed,  and  in  the  secondary 
a  corresponding  amount  of  power  is  produced. 

Since  the  same  magnetic  circuit  is  interlinked  with  both 
electric  circuits,  the  E.M.F.  induced  per  turn  must  be  the 
same  in  the  secondary  as  in  the  primary  circuit  ;  hence, 
the  primary  induced  E.M.F.  being  approximately  equal  to 
the  impressed  E.M.F.,  the  E.M.Fs.  at  primary  and  at  sec- 
ondary terminals  have  .approximately  the  ratio  of  their 
respective  turns.  Since  the  power  produced  in  the  second- 
ary is  approximately  the  same  as  that  consumed  in  the 
primary,  the  primary  and  secondary  currents  are  approxi- 
mately in  inverse  ratio  to  the  turns. 

117.  Besides  the  magnetic   flux  interlinked  with  both 
electric  circuits  —  which  flux,  in  a  closed  magnetic  circuit 
transformer,  has  a  circuit  of  low  reluctance  —  a  magnetic 
cross-flux  passes  between  the  primary  and  secondary  coils, 
surrounding  one  coil  only,  without  being  interlinked  with 
the  other.     This  magnetic  cross-flux  is  proportional  to  the 
current  flowing  in  the  electric  circuit,  or  rather,  the  ampere- 
turns  or  M.M.F.  increase  with  the  increasing  load  on  the 
transformer,  and   constitute  what   is  called   the   self-induc- 
tance of  the  transformer;  while  the  flux  surrounding  both 


168  ALTERNATING-CURRENT  PHENOMENA.       [§118 

coils  may  be  considered  as  mutual  inductance.  This  cross- 
flux  of  self-induction  does  not  induce  E.M.F.  in  the  second- 
ary circuit,  and  is  thus,  in  general,  objectionable,  by  causing 
a  drop  of  voltage  and  a  decrease  of  output  ;  and,  therefore, 
in  the  constant  potential  transformer  the  primary  and  sec- 
ondary coils  are  brought  as  near  together  as  possible,  or 
even  interspersed,  to  reduce  the  cross-flux. 

As  will  be  seen,  by  the  self-inductance  of  a  circuit,  not 
the  total  flux  produced  by,  and  interlinked  with,  the  circuit 
is  understood,  but  only  that  (usually  small)  part  of  the  flux 
which  surrounds  one  circuit  without  interlinking  with  the 
other  circuit. 

118.  The  alternating  magnetic  flux  of  the  magnetic 
circuit  surrounding  both  electric  circuits  is  produced  by 
the  combined  magnetizing  action  of  the  primary  and  of  the 
secondary  current. 

This  magnetic  flux  is  determined  by  the  E.M.F.  of  the 
transformer,  by  the  number  of  turns,  and  by  the  frequency. 

If 

<!>  =  maximum  magnetic  flux, 

N=  frequency,         •*,  - 

n  =  number  of  turns  of  the  coil  ; 

the  E.M.F.  induced  in  this  coil  is 


E  =      27r7Y^<l>10-8  volts, 
=  4.447W/$10-8  volts; 

hence,  if  the  E.M.F.,  frequency,  and  number  of  turns  are 
determined,  the  maximum  magnetic  flux  is 


To  produce  the  magnetism,  $,  of  the  transformer,  a 
M.M.F.  of  <5  ampere-turns  is  required,  which  is  determined 
by  the  shape  and  the  magnetic  characteristic  of  the  iron,  in 
the  manner  discussed  in  Chapter  X. 


§119]      ALTERNATING-CURRENT  TRANSFORMER.  169 

For  instance,  in  the  closed  magnetic  circuit  transformer, 
the  maximum  magnetic  induction  is 

1  "4   '  ? 

«j, 

where  5  =  the  cross-section  o'F  magnetic  circuit. 

119.  To  induce  a  magnetic  density,  <B,  a  M.M.F.  of  5C^ 
ampere-turns  maximum  is  required,  or,  3€m  /  V2  ampere- 
turns  effective,  per  unit  length  of  the  magnetic  circuit ; 
hence,  for  the  total  magnetic  circuit,  of  length,  /, 

7  TC 
SF  =  — —  ampere-turns  ; 

V2 

or  T       S       /3Cm  ~ 

/  =  —  =  - — 2L-  amps.  eff., 

«       «V2 
where  ^  =  number  of  turns. 

At  no  load,  or  open  secondary  circuit,  this  M.M.F.,  JF,  is 
furnished  by  the  exciting  current,  I00 ,  improperly  called  the 
leakage  •  current,  of  the  transformer ;  that  is,  that  small 
amount  of  primary  current  which  passes  through  the  trans- 
former at  open  secondary  circuit. 

In  a  transformer  with  open  magnetic  circuit,  such  as 
the  "hedgehog"  transformer,  the  M.M.F.,  SF,  is  the  sum 
of  the  M.M.F.  consumed  in  the  iron  and  in  the  air  part  of 
the  magnetic  circuit  (see  Chapter  X.). 

The  energy  of  the  exciting  current  is  the  energy  con- 
sumed by  hysteresis  and  eddy  currents  and  the  small  ohmic 
loss. 

The  exciting  current  is  not  a  sine  wave,  but  is,  at  least 
in  the  closed  magnetic  circuit  transformer,  greatly  distorted 
by  hysteresis,  though  less  so  in  the  open  magnetic  circuit 
transformer.  It  can,  however,  be  represented  by  an  equiv- 
alent sine  wave,  700,  of  equal  intensity  and  equal  power  with 
the  distorted  wave,  and  a  wattless  higher  harmonic,  mainly 
of  triple  frequency. 

Since  the  higher  harmonic  is  small  compared  with  the 


170 


ALTERNA  TING-CURRENT  PHENOMENA. 


12O 


total  exciting  current,  and  the  exciting  current  is  only  a 
small  part  of  the  total  primary  current,  the  higher  harmonic 
can,  for  most  practical  cases,  be  neglected,  and  the  exciting 
current  represented  by  the  equivalent  sine  wave. 

This  equivalent  sine  wave,  I00,  leads  the  wave  of  mag- 
netism, 3>,  by  an  angle,  a,  the  angle  of  hysteretic  advance  of 
phase,  and  consists  of  two  components,  —  the  hysteretic 
energy  current,  in  quadrature  with  the  magnetic  flux,  and 
therefore  in  phase  with  the  induced  E.M.F.  =  I00  sin  a;  and 
the  magnetizing  current,  in  phase  with  the  magnetic  flux, 
and  therefore  in  quadrature  with  the  induced  E.M.F.,  and 
so  wattless,  =  I00  cos  a. 

The  exciting  current,  700,  is  determined  from  the  shape 
and  magnetic  characteristic  of  the  iron,  and  number  of 
turns  ;  the  hysteretic  energy  current  is  — 

T     .             Power  consumed  in  the  iron 
J.nn  sin  a  = — . 

Induced  E.M.F. 

120.  Graphically,  the  polar  diagram  of  M.M.Fs.  ot  a 
transformer  is  constructed  thus  : 


Let,  in  Fig.  87,  O<&  =  the  magnetic  flux  in  intensity  and 
phase  (for  convenience,  as  intensities,  the  effective  values 
are  used  throughout),  assuming  its  phase  as  the  vertical ; 


§  12O]      ALTERNATING-CURRENT   TRANSFORMER.  171 

that  is,  counting  the  time  from  the  moment  where  the 
rising  magnetism  passes  its  zero  value. 

Then  the  resultant  M.M.F.  is  represented  by  the  vector 
O$,  leading  O<$>  by  the  anglfe  $O3>  =  03. 

The  induced  E.M..Fs.  have^the  phase  180°,  that  is,  are 
plotted  towards  the  left,  and  represented  by  the  vectors 
and 


If,  now,  wj'  =  angle  of  lag  in  the  secondary  circuit,  due 
to  the  total  (internal  and  external)  secondary  reactance,  the 
secondary  current  7t  ,  and  hence  the  secondary  M.M.F., 
•9r1=  n^  /x,  will  lag  behind  E±  by  an  angle  ft',  and  have  the 
phase,  180°  -f  /?',  represented  by  the  vector  O^l.  Con- 
structing a  parallelogram  of  M.M.Fs.,  with  O$  as  a  diag- 
onal and  O&i  as  one  side,  the  other  side  or  O$0  is  the 
primary  M.M.F.,  in  intensity  and  phase,  and  hence,  dividing 
by  the  number  of  primary  turns,  n0  ,  the  primary  current  is 

Io  =  $ol  no  • 

To  complete  the  diagram  of  E.M.Fs.  ,  we  have  now,  — 
In  the  primary  circuit  : 

E.M.F.  consumed  by  resistance  is  I0rot  in  phase  with  I0,  and 
represented  by  the  vector  OEor  ; 

E.M.F.  consumed  by  reactance  is  I0x0,  90°  ahead  of  I0  ,  and 
represented  by  the  vector  OE0  x  ; 

E.M.F.  consumed  by  induced  E.M.F.  is  £0,  equal  and  oppo- 
site thereto,  and  represented  by  the  vector  OE0f  • 

Hence,  the  total  primary  impressed  E.M.F.  by  combina- 
tion of  OEor,  OEOK,  and  OEJ1  by  means  of  the  parallelo- 
gram of  E.M.Fs.  is, 

£0=  OE0, 

and  the  difference  of  phase  between  the  primary  impressed 
E.M.F.  and  the  primary  current  is 

ft  =  E0  O$0. 

"  In  the  secondary  circuit  : 

Counter  E.M.F.  of  resistance  is  /i^  in  opposition  \vith/j, 
and  represented  by  the  vector  OE±  r  •  k 


172 


ALTERNATING-CURRENT  PHENOMENA. 


121 


Counter  E.M.F.  of  reactance  is  /^i,  90°  behind  flt  and 
represented  by  the  vector  OE\K'  ; 

Induced  E.M.Fs.,  E{  represented  by  the  vector  OE{. 

Hence,  the  secondary  terminal  voltage,  by  combination 
of  OE^y  OE^  and  OE±  by  means  of  the  parallelogram  of 
E.M.Fs.  is  £!  =  OEly 

and  the  difference  of  phase  between  the  secondary  terminal 
voltage  and  the  secondary  current  is 


As  will  be  seen  in  the  primary  circuit  the  "components 
of  impressed  E.M.F.  required  to  overcome  the  counter 
E.M.Fs."  were  used  for  convenience,  and  in  the  secondary 
circuit  the  "counter  E.M.Fs." 


Fig.  88.     Transformer  Diagram  with  80°  Lag  in  Secondary  Circuit. 

121.  In  the  construction  of  the  transformer  diagram,  it. 
is  usually  preferable  not  to  plot  the  secondary  quantities, 
current  and  E.M.F.,  direct,  but  to  reduce  them  to  corre- 
spondence with  the  primary  circuit  by  multiplying  by  the 
ratio  of  turns, — a  =  n0/  nv  for  the  reason  that  frequently 
primary  and  secondary  E.M.Fs.,  etc.,  are  of  such  different 


§121]        ALTERNATING-CURRENT  TRANSFORMER. 


173 


magnitude  as  not  to  be  easily  represented  on  the  same 
scale;  or  the  primary  circuit  may  be  reduced  to  the  sec- 
ondary in  the  same  way.  In  .either  case,  the  vectors  repre- 
senting the  two  induced  E.lVl.Fs.  coincide,  or 


Fig.  89.     Transformer  Diagram  with  50"  Lag  in  Secondary  Circuit. 

Figs.  88  to  94  give  the  polar  diagram  of  a  transformer 
having  the  constants  — 

r0  =  .2  ohms,  b0    =  .0173  mhos, 

x0  =  .33  ohms,  Ejf  =  100  volts, 

r±  =  .00167  ohms,  /i     =  60  amperes, 

x1  =  .0025  ohms,  a     =  10. 
g0  =  .0100  mhos, 

for  the  conditions  of  secondary  circuit, 


S  =  80°  lag  in  Fig.  88. 

50°  lag  "        89. 

20°  lag  «        90. 

O,  -or  in  phase,  "        91. 


ft'  =  20P  lead  in  Fig.  92, 
50°  lead  "  93. 
80°  lead  "  94. 


As  shown  with  a  change  of  /?/,  E0,gl,g0,  etc.,  change 
in  intensity  and  direction.  The  locus  described  by  them 
are  circles,  and  are  shown  in  Fig.  95,  with  the  point  corre- 
sponding to  non-inductive  load  marked.  The  part  of  the 
locus  corresponding  to  a  lagging  secondary  current  is 


174  ALTERNATING-CURRENT  PHENOMENA.         [§121 


Fig.  90.    Transformer  Diagram  with  20°  Lag  in  Secondary  Circuit. 


Eo 


Fig.  91.     Transformer  Diagram  with  Secondary  Current  in  Phase  with  E.M.F. 


Fig.  92.     Transformer  Diagram  with  20°  Lead  in  Secondary  Current. 


§121]     ALTERNATING-CURRENT  TRANSFORMER.  175 


E, 


E,' 


Fig.  93.     Transformer  Diagram  with  50°  Lead  in  Secondary  Circuit 


Fig.  94.     Transformer  Diagram  with  80°  Lead  in  Secondary  Circuit. 


Fig.  95. 


176 


ALTERNATING-CURRENT  PHENOMENA.       [§  122 


shown  in  thick  full   lines,  and  the  part   corresponding   to 
leading  current  in  thin  full  lines. 

122.  This  diagram  represents  the  condition  of  con- 
stant secondary  induced  E.M.F.,  E^,  that  is,  corresponding 
to  a  constant  maximum  magnetic  flux. 

By  changing  all  the  quantities  proportionally  from  the 
diagram  of  Fig.  95,  the  diagram  for  the  constant  primary 
impressed  E.M.F.  (Fig.  96),  and  for  constant  secondary  ter- 
minal voltage  (Fig.  97),  are  derived.  In  these  cases,  the 
locus  gives  curves  of  higher  order. 


Fig.  96. 


Fig.  98  gives  the  locus  of  the  various  quantities  when 
the  load  is  changed  from  fulLload,  7X  =  60  amperes  in  a 
non-inductive  secondary  external  circuit  to  no  load  or  open 
circuit. 

a.)  By  increase  of  secondary  resistance ;  b.)  by  increase 
of  secondary  inductive  reactance  ;  c.}  by  increase  of  sec- 
ondary capacity  reactance. 

As  shown  in  a.},  the  locus  of  the  secondary  terminal  vol- 
tage, E19  and  thus  of  E0,  etc.,  are  straight  lines;  and  in 
b.)  and  c.},  parts  of  one  and  the  same  circle  a.)  is  shown 


§  123]      ALTERNATING-CURRENT  TRANSFORMER.  177 

in  full  lines,  b.)  in  heavy  full  lines,  and  c.)  in  light  fiill  lines. 
This  diagram  corresponds  to  constant  maximum  magnetic 
flux ;  that  is,  to  constant  secondary  induced  E.M.F.  The 
diagrams  representing  constant  primary  impressed  E.M.F. 
and  constant  secondary  terminal  voltage  can  be  derived 
from  the  above  by  proportionality. 


Fig.  97. 


123.  It  must  be  understood,  however,  that  for  the  pur- 
pose of  making  the  diagrams  plainer,  by  bringing  the  dif- 
ferent values  to  somewhat  nearer  the  same  magnitude,  the 
constants  chosen  for  these  diagrams  represent,  not  the  mag- 
nitudes found  in  actual  transformers,  but  refer  to  greatly 
exaggerated  internal  losses. 

In  practice,  about  the  following  magnitudes  would  be 
found  : 


r0  =  .01 
x0  =  .033 


ohms; 
ohms ; 


=  .00008  ohms ; 


Xi  =  .00025  ohms  ; 
g0  =  .001  ohms  ; 
b0  =  .00173  ohms  ; 


that   is,  about  one-tenth  as  large  as  assumed.     Thus  the 

E-. ,  etc.,  under  the  different 


changes  of  the  values  of  E0, 


conditions  will  be  very  much  smaller. 


178 


ALTERNATING-CURRENT  PHENOMENA.       [§124 


Symbolic  Method. 

124.  In  symbolic  representation  by  complex  quantities 
the.  transformer  problem  appears  as  follows  : 

The  exciting  current,  700,  of  the  transformer  depends 
upon  the  primary  E.M.F.,  which  dependance  can  be  rep- 
resented by  an  admittance,  the  "  primary  admittance," 
Y0=.  g0-\-jboJ  of  the  transformer. 


Fig.  98. 

The  resistance  and  reactance  of  the  primary  and  the 
secondary  circuit  are  represented  in  the  impedance  by 

Z0  =  r0  —  jx0,         and         Zl  =  r±  —  j  x^. 

Within  the  limited  range  of  variation  of  the  magnetic 
density  in  a  constant  potential  transformer,  admittance  and 
impedance  can  usually,  and  with  sufficient  exactness,  be 
considered  as  constant. 

Let 

n0   =  number  of  primary  turns  in  series ; 
«i    =  number  of  secondary  turns  in  series; 
a     =  —  =  ratio  of  turns ; 

Y0  =  go  —  jb0  =  primary  admittance 

Exciting  current 

Primary  counter  E.M.F. ' 


§  124]      ALTERNATING-CURRENT  TRANSFORMER. 
Z0  =  r0  —  j XQ  =  primary  impedance 

E.M.F.  consumed  in  primary  coil  by  resistance  and  reactance. 

Primary  current 

Z1  =  ri  —  j x±  =  secondary  impedance 

E.M.F.  consumed  in  sedbndary  coil  by  resistance  and  reactance  . 

Secondary  current 

where  the  reactances,  x0  and  x^ ,  refer  to  the  true  self -induc- 
tance only,  or  to  the  cross-flux  passing  between  primary  and 
secondary  coils  ;  that  is,  interlinked  with  one  coil  only. 
Let  also 

Y    —g^jb  —  total    admittance    of   secondary   circuit, 

including  the  internal  impedance  ; 
E0  =  primary  impressed  E.M.F. ; 
E0'  =  E.M.F.  consumed  by  primary  counter  E.M.F. ; 
EI.  =  secondary  terminal  voltage; 
EI   =  secondary  induced  E.M.F. ; 
I0    =  primary  current,  total ; 
I00   =  primary  exciting  current ; 
^     —  secondary  current. 

Since  the  primary  counter  E.M.F.,  EJ,  and  the  second- 
ary induced  E.M.F.,  E^,  are  proportional  by  the  ratio  of 
turns,  a, 

£.'  =   -  *£{.  (1) 

The  secondary  current  is  : 

7,     =  KE/,  (2) 

consisting  of  an  energy  component,  gE^,  and  a  reactive 
component,  g  E-[. 

To  this  secondary  current  corresponds  the  component  of 
primary  current, 

/  f 


The  primary  exciting  current  is  — 

f00=y0Ej.  (4) 

Hence,  the  total  primary  current  is : 


180  ALTERNATING-CURRENT  PHENOMENA.        [§  125 


or, 


The  E.M.F.  consumed  in  the  secondary  coil  by  the 
internal  impedance  is  Z^I^. 

The  E.M.F.  induced  in  the  secondary  coil  by  the  mag- 
netic flux  is  E-[. 

Therefore,  the  secondary  terminal  voltage  is 


or,  substituting  (2),  we  have 

The  E.M.F.  consumed  in  the  primary  coil  by  the  inter- 
nal impedance  is  Z0 10. 

The  E.M.F.  consumed  in  the  primary  coil  by  the  counter 
E.M.F.  is  El. 

Therefore,  the  primary  impressed  E.M.F.  is 

or,  substituting  (6), 

"*  Z  Y  *  (8) 

•  +  Z.y.  +  ^? 


125.    We  thus  have, 

primary  E.M.F.,      E0  =  -  aE{  j  1  +  Z0  Y0  +  ^1  j  ,  (8) 

secondary  E.M.F.,  El  =  E{  { 1  -  Z,  Y},  (7) 

primary  current,      f0  =  -  ^  {  Y  -f  a2  Y0 } ,  (6) 

a 

secondary  current,  7T   =  YE{,  (2) 

as  functions  of  the  secondary  induced  E.M.F.,  E^  as  pa- 
rameter. 


§125]       ALTERNATING-CURRENT  TRANSFORMER.  181 

From  the  above  we  derive 

Ratio  of  transformation  of  E.M.Fs.  : 


(9) 


Ratio  of  transformations  of  currents  : 


From    this    we     get,    at     constant     primary    impressed 
E.M.F., 

E0  =  constant  ; 

secondary  induced  E.M.F., 


a 


a- 
E.M.F.  induced  per  turn, 


a 


secondary  terminal  voltage, 


primary  current, 


E0 


secondary  current, 

77  \r 

T  -fiin  -I 


At  constant  secondary  terminal  voltage, 
EI  =  const. ; 


182  ALTERNATING-CURRENT  PHENOMENA.        [§126 

secondary  induced  E.M.F., 


E.M.F.  induced  per  turn, 


*  l-Z,y' 

primary  impressed  E.M.F., 


E0     =  —  aE^ 


1  -  Z^ 

primary  current, 

El 

;1 

secondary  current, 


(12) 


126.    Some  interesting  conclusions  can  be  drawn  from 
these  equations. 

The  apparent  impedance  of  the  total  transformer  is 


J0 


(13) 


Substituting  now,    —  =  V,  the  total  secondary  admit- 
tance, reduced  to  the  primary  circuit  by  the  ratio  of  turns, 

' 


Y0-\-  Y1  is  the  total  admittance  of  a  divided  circuit  with 
the  exciting  current,  of  admittance  Y0,  and  the  secondary 


§127]     ALTERNATING-CURRENT  TRANSFORMER. 


183 


current,  of  admittance  Y'  (reduced  to  primary),  as  branches. 
Thus  : 


is  the  impedance  of  this  divided  circuit,  and 

Z0  =  Z0'  +  Z0. 
That,  is  :     . 


(17) 


The  alternate-current  transformer,  of  primary  admittance 
Y0  ,  total  secondary  admittance  V,  and  primary  impedance 
Z0,  is  equivalent  to,.  and  can  be  replaced  by,  a  divided  circuit 
with  the  branches  of  admittance  V0,  the  exciting  current,  and 
admittance  Y'  =  Y/a2  ,  the  secondary  current,  fed  over  mains 
of  the  impedance  Z0  ,  tJie  internal  primary  impedance. 

This  is  shown  diagrammatically  in  Fig.  99. 


Generator 


Transformer 


I 

E. 


Receiving: 
Circuit 


I 


Fig.  99. 

127.  Separating  now  the  internal  secondary  impedance 
from  the  external  secondary  impedance,  or  the  impedance  of 
the  consumer  circuit,  it  is 

1 


where  Z  =  external  secondary  impedance, 


(18) 
(19) 


184  ALTERNATING-CURRENT  PHENOMENA.        [§127 

Reduced  to  primary  circuit,  it  is 


That  is : 


Y 
=  Z/  +  Z'.  (20) 


An  alternate-current  transformer,  of  primary  admittance 
Y0,  primary  impedance  Z0,  secondary  impedance  Z^,  and 
ratio  of  turns  a,  can,  when  the  secondary  circtiit  is  closed  by 
an  impedance  Z  (the  impedance  of  the  receiver  circuit],  be 
replaced,  and  is  equivalent  to  a  circuit  of  impedance  Z '  = 
T,  fed  over  mains  of  the  impedance  Z0-\-  Z^,  where  Z^  = 
,  shunted  by  a  circuit  of  admittance  Y0,  which  latter 
circuit  branches  off  at  the  points  a  —  b,  between  the  impe- 
dances Zn  and  Zl . 


Generator  Transformer 


Receiving 
Circuit  • 

7. 


Zo       cJ      Zfa2z,          Q) 

yJ  fz-Vz 

\ 1  * 


b 
Fig.  100. 

This  is  represented  diagrammatically  in  Fig.  100. 

It  is  obvious  therefore,  that  if  the  transformer  contains 
several  independent  secondary  circuits  they  are  to  be  con- 
sidered as  branched  off  at  the  points  a  —  b,  in  diagram 
Fig.  100,  as  shown  in  diagram  Fig.  101. 

It  therefore  follows : 

An  alternate-ctirrent  transformer,  of  x  secondary  coils,  of 
the  internal  impedances  Z^,  Z^1,  .  .  .  Z-f,  closed  by  external 
secondary  circuits  of  the  impedances  Z1,  Z11,  .  .  .  Zx,  is  equiv- 
alent to  a  divided  circuit  of  x  -f-  1  branches,  one  branch  of 


§127]       AL  TERN  A  TING-CURRENT  TRANSFORMER. 


185 


Generator 


Transformer 


Receiving 
Circuits 

Z1 


Fig.  101. 

admittance  Y0,  the  exciting  current,  the  other  branches  of  the 
impedances  Z/  +  Z1,  Z^n  +  Zn,  .  .  .  Zf  +  Z*  the  latter 
impedances  being  reduced  to  the  primary  circuit  by  tJie  ratio 
of  tiirns,  and  the  whole  divided  circuit  being  fed  by  the 
primary  impressed  E.M.F.  E0,  over  mains  of  the  impedance 

z. 

Consequently,  transformation  of  a  circuit  merely  changes 
all  the  quantities  proportionally,  introduces  in  the  mains  the 
impedance  Z0  -f  Z^,  and  a  branch  circuit  between  Z0  and 
Z±,  of  admittance  Y0. 

Thus,  double  transformation  will  be  represented  by  dia- 
gram, Fig.  102. 

Transformer  Transformer 


Receiving 
Circuits 


z0 


UJ) 


Yoc 


Fig.  102. 


183  ALTERNATING-CURRENT  PHENOMENA.       [§128 

With  this  the  discussion,  of  the  alternate-current  trans- 
former ends,  by  becoming  identical  with  that  of  a  divided 
circuit  containing  resistances  and  reactances. 

Such  circuits  have  explicitly  been  discussed  in  Chapter 
VIII.,  and  the  results  derived  there  are  now  directly  appli- 
cable to  the  transformer,  giving  the  variation  and  the  con- 
trol of  secondary  terminal  voltage,  resonance  phenomena,  etc. 

Thus,  for  instance,  if  Z^  =  Z0,  and  the  transformer  con- 
tains a  secondary  coil,  constantly  closed  by  a  condenser 
reactance  of;  such  size  that  this  auxiliary  circuit,  together 
with  the  exciting  circuit,  gives  the  reactance  —x0,  with  a 
non-inductive  secondary  circuit  Zl  =  1\,  we  get  the  condi- 
tion of  transformation  from  constant  primary  potential  to 
constant  secondary  current,  and  inversely,  as  previously 
discussed. 

Non-inductive  Secondary  Circuit. 

128.    In  a  non-inductive  secondary  circuit,  the  external 
secondary  impedance  is, 

Z  =  Ri, 
or,  reduced  to  primary  circuit, 

Z  -  ^  -   p 

^~^~ 

Assuming  the   secondary   impedance,   reduced  to   primary 
circuit,  as  equal  to  the  primary  impedance, 


0  J   XQ 

itis' 


Substituting  these  values  in  Equations  (9),  (10),  and  (13), 
we  have 

Ratio  of  E.M.Fs.  : 


r0—jx0 


§  128]       ALTERNATING-CURRENT  TRANSFORMER.  187 


. 
K  -f-  r0      j  x0 

,         r0  —  jx0  f      r0—jx0 

r 


or,  expanding,  and  neglecting  terms  of  higher  than  third 
order, 


R 


or,  expanded, 


Neglecting  terms  of  tertiary  order  also, 

—  °  =±  —  ^  |  1  +  2 
£i  ( 

Ratio  of  currents  : 


or,  expanded, 


Neglecting  terms  of  tertiary  order  also, 


Total  apparent  primary  admittance  : 

1  +        r°-jX\      +  (r0  -jx0}  (go 


•          r0  —  Jx0 
{R  +  (r0  -Jx0)  +  R  (r0-jx^  (g0  +/^)}  {1-  (g0 


(r0  -  jx0)  -  ^  (g0  +jb0)-1R  ( 


r 


188  ALTERNATING-CURRENT  PHENOMENA.        [§129 

or, 


Neglecting  terms  of  tertiary  order  also  : 
<  =  R     1  +  2          /£o  _R  (go  +  , 


Angle  of  lag  in  primary  circuit  : 
tan  o>0  =  J  ?  ,  hence, 


2  r0b0  - 
tan  £    =  --  —  — 


1  +  ~  -  -  £&  ~  2  rogo  -2x0b0  +  &g*  +  JR* 

Neglecting  terms  of  tertiary  order  also  : 
-£24.  jf^j 


tan  <o0  = 


129.  If,  now,  we  represent  the  external  resistance  of 
the  secondary  circuit  at  full  load  (reduced  to  the  primary 
circuit)  by  R0,  and  denote, 


r0    _        __        f-  Internal  resistance  of  transformer  _  percentage 

R0     '  External  resistance  of  secondary  circuit  ~  nal  resistance, 

2  X0   __        __  ra^jQ       Internal  reactance  of  transformer          __  percentage 
J£  External  resistance  of  secondary  circuit  I\2i\  reactance 

*•*-  h  =  ratio  =  percentage  hysteresis,' 

Magnetizing  current         percentage  magnetizing  cur- 

-  -  -  • 

rent 


—  r,  - 

n  "0  -    M    -  :  -  • 

1  otal  secondary  current 


and  if  d  represents  the  load  of  the  transformer,  as  fraction 
of  full  load,  we  have 


§  129]      ALTERNATING-CURRENT  TRANSFORMER.  189 

and,  ^=P^ 

K 

^q4 
R  > 

B  h 

Xg.  -  ^, 

**.-* 

a 

Substituting  these  values  we  get,  as  the  equations  of  the 
transformer  on  non-inductive  load, 
Ratio  of  E.M.Fs.  : 


\  1  +  rf(p  -/q) 

( 


or,  eliminating  imaginary  quantities, 


Ratio  of  currents  : 

LL=        1  fi  ...    (h  +/g)    ,   (P  -y 
/i  «(  </ 


or,  eliminating  imaginary  quantities, 


.     1    , 


190  AL  TERNA  TING-CURRENT  PHENOMENA.       [§129 

Total  apparent  primary  impedance  : 


Z,  - 


or,  eliminating  imaginary  quantities, 


, 


Angle  of  lag  in  primary  circuit : 

dq  4~      +  Pg  —  gh  —  2  — ^ 

tan  <o0  =  — 

.          ,          h  ,                 .    hg  h2 

J_  -\-  «p  — pn  —  Cjg  -[-  — ^*  — 

d  d 


That  is, 

An  alternate-current  transformer,  feeding  into  a  non-induc- 
tive secondary  circuit,  is  represented  by  the  constants : 

_R0  =  secondary  external  resistance  at  full  load; 

p    =  percentage  resistance  ; 

q     =  percentage  reactance  ; 

h     =  percentage  hysteresis  ; 

g    =  percentage  magnetizing  current ; 

d    =  secondary  percentage  load. 

All  these  qualities  being  considered  as  reduced  to  the  primary 
circuit  by  the  sqtiare  of  the  ratio  of  turns,  a2. 


§130]      ALTERNATING-CURRENT   TRANS-FORMER.  191 

-      130.    As   an  instance,  a  transformer   of   the  following 
constants  may  be  given : 


e0    =1,000; 
a     =        10; 


>X0=     120; 
P,  =-02- 


Substituting  these  values,  gives  : 

100 


h  .=  .02 ; 
g  =  .04. 


V(1.0014  +  .02  d^f  +  (.0002  +  .06  d)* 


1.0014  +          --.  0002; 


=  .  1  ^ 


.06  ^  +       --. 


tan  Co,,  = 


Fig.  103.     Load  Diagram  of  Transformer. 


192  ALTERNATING-CURRENT  PHENOMENA.         [§13(> 

In  diagram  Fig.  103  are  shown,  for  the  values  from 
d  =  0  to  d=  1.5,  with  the  secondary  current  c±  as  abscis- 
sae, the  values : 

secondary  terminal  voltage,  in  volts, 

secondary  drop  of  voltage,  in  per  cent, 

primary  current,  in  amps, 

excess    of    primary    current    over    proportionality    with 

secondary,  in  per  cent, 
primary  angle  of  lag. 

The  power-factor  of  the  transformer,  cos  <o0,  is  .45  at 
open  secondary  circuit,  and  is  above  .99  from  25  amperes, 
upwards,  with  a  maximum  of  .995  at  full  load. 


ALTERNATING-CURRENT  TRANSFORMER.  193 


* 

CHAPTER    XIV. 

THE    GENERAL    ALTERNATING-CURRENT    TRANSFORMER. 

131.  THE  simplest  alternating-current  apparatus  is  the 
alternating-current  transformer.  It  consists  of  a  magnetic 
circuit,  interlinked  with  two  electric  circuits  or  sets  of 
electric  circuits.  The  one,  the  primary  circuit,  is  excited 
by  an  impressed  E.M.F.,  while  in  the  other,  the  secondary 
circuit,  an  E.M.F.  is  induced.  Thus,  in  the  primary  circuit, 
power  is  consumed,  in  the  secondary  circuit  a  correspond- 
ing amount  of  power  produced  ;  or  in  other  words,  power 
is  transferred  through  space,  from  primary  to  secondary 
circuit.  This  transfer  of  power  finds  its  mechanical  equiv- 
alent in  a  repulsive  thrust  acting  between  primary  and 
secondary.  Thus,  if  the  secondary  coil  is  not  held  rigidly 
as  in  the  stationary  transformer,  it  will  be  repelled  and 
move  away  from  the  primary.  This  mechanical  effect  is 
made  use  of  in  the  induction  motor,  which  'represents  a 
transformer  whose  secondary  is  mounted  movably  with  re- 
gard to  the  primary  in  such  a  way  that,  while  set  in  rota- 
tion, it  still  remains  in  the  primary  field  of  force. 

The  condition  that  the  secondary  circuit,  while  moving 
with  regard  to  the  primary,  does  not  leave  the  primary  field 
of  magnetic  force,  requires  that  this  field  is  not  undirec- 
tional,  but  that  an  active  field  exists  in  every  direction. 
One  way  of  producing  such  a  magnetic  field  is  by  exciting 
different  primary  circuits  angularly  displaced  in  space  with 
each  other  by  currents  of  different  phase.  Another  way  is 
to  excite  the  primary  field  in  one  direction  only,  and  get 
the  cross  magnetization,  or  the  angularly  displaced  mag- 
netic field,  by  the  reaction  of  the  secondary  current. 


194  A  L  TEJtNA  TING-  C  URRENT  PHENOMENA .  .       [§131 

We  see,  consequently,  that  the  stationary  transformer 
and  the  induction  motor  are  merely  different  applications 
of  the  same  apparatus,  comprising  a  magnetic  circuit  in- 
terlinked with  two  electric  circuits.  Such  an  apparatus 
can  properly  be  called  a  "general  alternating- current  trans- 
former}' 

The  equations  of  the  stationary  transformer  and  those 
of  the  induction  motor  are  merely  specializations  of  the 
general  alternating-current  transformer  equations. 

Quantitatively  the  main  differences  between  induction 
motor  and  stationary  transformer  are  those  produced  by 
the  air-gap  between  primary  and  secondary,  which  is  re- 
quired to  give  the  secondary  mechanical  movability.  This 
air-gap  greatly  increases  the  magnetizing  current  over  that 
in  the  closed  magnetic  circuit  transformer,  and  requires 
an  ironclad  construction  of  primary  and  secondary  to  keep 
the  magnetizing  current  within  reasonable  limits.  An  iron- 
clad construction  again  greatly  increases  the  self-induction 
of  primary  and  secondary  circuit.  Thus  the  induction; 
motor  is  a  transformer  of  large  magnetizing  current  and 
large  self-induction  ;  that  is,  comparatively  large  primary 
susceptance  and  large  reactance. 

The  general  alternating-current  transformer  transforms 
between  electrical  and  mechanical  power,  and  changes  not 
only  E.M.Fs.  and  currents,  but  frequencies  also. 

132.  Besides  the  magnetic  flux  interlinked  with  both 
primary  and  secondary  electric  circuit,  a  magnetic  cross- 
flux  passes  in  the  transformer  between  primary  and  second- 
ary, surrounding  one  coil  only,  without  being  interlinked 
with  the  other.  This  magnetic  cross-flux  is  proportional  to 
the  current  flowing  in  the  electric  circuit,  and  constitutes 
what  is  called  the  self-induction  of  the  transformer.  As 
seen,  as  self-induction  of  a  transformer  circuit,  not  the  total 
flux  produced  by  and  interlinked  with  this  circuit  is  under- 
stood, but  only  that  —  usually  small  —  part  of  the  flux 


§§133-135]    ALTERNATIATG-CURRENT  TRANSFORMER.    195 

which  surrounds  the  one  circuit  without  interlinking  with 
the'  other,  and  is  thus  produced  by  the  M.M.F.  of  one: 
circuit  only. 

j 
133.    The  common  magnetic  flux  of  the  transformer  is 

produced  by  the  resultant  M.M.F.  of  both  electric  circuits. 
It  is  determined  by  the  counter  E.M.F.,  the  number  of 
turns,  and  the  frequency  of  the  electric  circuit,  by  the 
equation  : 


Where  E  =  effective  E.M.F. 

A^=  frequency. 
n    =  number  of  turns. 
<£  =  maximum  magnetic  flux. 

The  M.M.F.  producing  this  flux,  or  the  resultant  M.M.F. 
of  primary  and  secondary  circuit,  is  determined  by  shape 
and  magnetic  characteristic  of  the  material  composing  the 
magnetic  circuit,  and  by  the  magnetic  induction.  At  open 
secondary  circuit,  this  M.M.F.  is  the  M.M.F.  of  the  primary 
current,  which  in  this  case  is  called  the  exciting  current, 
and  consists  of  an  energy  component,  the  magnetic  energy 
current,  and  a  reactive  component,  the  magnetizing  current. 

134.  In    the   general    alternating-current  -transformer, 
where  the  secondary  is  movable  with  regard  to  the  primary, 
the  rate  of  cutting  of  the  secondary  electric  circuit  with  the 
mutual  magnetic  flux  is  different  from  that  of  the  primary. 
Thus,  the  frequencies  of  both  circuits  are  different,  and  the 
induced   E.M.Fs.    are   not   proportional   to   the   number  of 
turns  as  in  the  stationary  transformer,  but  to  the  product 
of  number  of  turns  into  frequency. 

135.  Let,  in  a  general  alternating-current  transformer: 

frequency,  or  "slip"; 


thus,  if 

JV=  primary  frequency,  or  frequency  of  impressed  E.M.F., 
sJV=  secondary  frequency  ; 


196  A  L  TERN  A  TING-CURRENT  PHENOMENA.        [§135 

and  the  E.M.F.  induced  per  secondary  turn  by  the  mutual 
flux  has  to  the  E.M.F.  induced  per  primary  turn  the  ratio  sy 

s  =  0  represents  synchronous  motion  of  the  secondary ; 

s  <  0  represents  motion  above  synchronism — driven  by  external 

mechanical  power,  as  will  be  seen ; 
s  =  1  represents  standstill ; 
s  >  1  represents  backward  motion  of  the  secondary 

that  is,  motion  against  the  mechanical  force  acting  between 
primary  and  secondary   (thus   representing  driving  by  ex- 
ternal mechanical  power). 
Let 

nQ   =  number  of  primary  turns  in  series  per  circuit ; 
MI    =  number  of  secondary  turns  in  series  per  circuit ; 

a     =  —  =  ratio  of  turns  ; 

«i 
YQ  =  gQ  -\-jbQ  —  primary  admittance  per  circuit ; 

where 

gQ    =  effective  conductance  ; 

£0    =  susceptance  ; 

ZQ '  =  ro  —  Jxo  =  internal  primary  impedance  per  circuit, 

where 

r0    =  effective  resistance  of  primary  circuit ; 

x0   =  reactance  of  primary  circuit ; 

Zn  =  /i  —  j'xi  =  internal  secondary  impedance  per  circuit 

at  standstill,  or  for  s  =  1, 
where 

ri    =  effective  resistance  of  secondary  coil ; 
Xi   =  reactance  of  secondary  coil  at  standstill,  or  full  fre- 
quency, s  =  1. 

Since  the  reactance  is  proportional  to  the  frequency,  at 
the  slip  s,  or  the  secondary  frequency  s  N,  the  secondary 

impedance  is  ; 

Zi  =  r\  —  Jsx\- 

Let  the  secondary  circuit  be  closed  by  an  external  re- 
sistance r,  and  an  external  reactance,  and  denote  the  latter 


$135]      ALTERNATING-CURRENT  TRANSFORMER.  197 

by  x  at  frequency  N>  then  at  frequency  s  N,  or  slip  s,  it 
will  be  =  s  x>  and  thus  : 

Z  =  r  —  jsx  =  external  -secondary  impedance.* 
Let  ^ 

EQ  —  primary  impressed  E.M.F.  per  circuit, 
EQ  =  E.M.F.  consumed  by  primary  counter  E.M.F., 
E^   =  secondary  terminal  E.M.F., 
E{  =  secondary  induced  E.M.F., 
e      =  E.M.F.  induced  per  turn  by  the  mutual  magnetic  flux, 

at  full  frequency  W, 
IQ    =  primary  current, 
fQO  =  primary  exciting  current, 
/!     =  secondary  current. 

It  is  then  : 

Secondary  induced  E.M.F. 

EI  =  sn^e. 

Total  secondary  impedance 

ZL  +  Z  =  (r,  +  r)  -js  (*,  +  x)  ; 

hence,  secondary  current 

_        EI        _  kn^e 


~  Zx  +  Z  ~  (r,  +  r)  -js  (Xl  +  x)  ' 
Secondary  terminal  voltage 


i  -,                   fi  —  /o^i              I              sn^e(r  —  / 's x) 
—  sn^e  \  1 — ^—  — ^ J- *- 


(r,  +  r)  -js  (x,  +  *>)  )        (rj  +  r)  -js  (Xl  +  x) 

*  This  applies  to  the  case  where  the  secondary  contains  inductive  reac- 
tance only  ;  or,  rather,  that  kind  of  reactance  which  is  proportional  to  the  fre- 
quency. In  a  condenser  the  reactance  is  inversely  proportional  to  the  frequency 
in  a  synchronous  motor  under  circumstances  independent  of  the  frequency. 
Thus,  in  general,  we  have  to  set,  x  —  x'  +  x"  -f  x'" ',  where  x'  is  that  part  of 
the  reactance  which  is  proportional  to  the  frequency,  x1'  that  part  of  the  reac- 
tance independent  of  the  frequency,  and  x'"  that  part  of  the  reactance  which 
is  inversely  proportional  to  the  frequency ;  and  have  thus,  at  slip  s,  or  frequency 
sNt  the  external  secondary  reactance  sx'  +  x"  +  — . 


198  ALTERNATING-CURRENT  PHENOMENA.        [§135 

E.M.F.  consumed  by  primary  counter  E.M.F. 

EQ      =     —    «0*  J 

hence,  primary  exciting  current : 

700  =  jE0'Y0  =  —  nQe(gQ  +  /£<>)• 

Component  of  primary  current  corresponding  to  second- 
ary current  II: 


tf<i  +  r)-     s 
hence,  total  primary  current, 

/o  =  ^oo  +  // 


Primary  impressed  E.M.F., 


We  get  thus,  as  the 
Equations  of  the  General  Alternating-Citrrent  Transformer  : 

Of  ratio  of  turns,  a  ;  and  ratio  of  frequencies,  s  ;  with  the 
E.M.F.  induced  per  turn  at  full  frequency,  e,  as  parameter, 
the  values  : 

Primary  impressed  E.M.F., 


Secondary  terminal  voltage, 

r^-jsx^  _       i  r-sx 


F  -    ™ 
^ 

Primary  current, 


§135]       ALTERNATING-CURRENT  TRANSFORMER.  19Q 

Secondary  current, 


Therefrom,  we  get : 
Ratio  of  currents, 
/o  _        1  (  -,    ,    *2  / 

-7,=  ~a\l  -y(< 

Ratio  of  E.M.Fs.-, 

.i  +  4 

±1° 
^i 


Total  apparent  primary  impedance, 


tf2  fa  +  r)  —  js  (xl  +  # 


where 


in  the  general  secondary  circuit  as  discussed  in  foot-note,. 
page  4. 

Substituting  in  these  equations  : 

i~i, 

gives  the 

General  Equations  of  the  Stationary  Alternating-Current 
Transformer  : 

,    Z0F0 


a2  Z±  +  Z 

27  1  Z\  } 

1  =  n*- 


70    =  —  n0 


+  Z) 


200  AL  TERNA  TING-CURRENT  PHENOMENA.       [§  1 35 


V 
_  <T  (Zi  -f 


zt  =     =  «.  (^ 


Substituting  in  the  equations  of  the  general  alternating- 
current  transformer, 

Z  =  0, 
gives  the 

General  Equations  of  the  Induction  Motor: 


±Jh 

s 


rl—jsxl 

-  -     +  -  (^  +  j 


2 

„  /  •       x  a  r±  —  j  s  xl 

Zt  =  —  (r,  -/- 


Returning  now  to  the  general  alternating-current  trans- 
former, we  have,  by  substituting 

(ri  +  r)2  +  s*  (Xl  +  ^)2  =  ^2? 

and  separating  the  real  and  imaginary  quantities, 
4-       —  (r0  (^  4-  r)  +  sxo  (*i  +  *)) 


§  136]      ALTERNATING-CURRENT  TRANSFORMER.  201 

+  J 


Neglecting  the  exciting  current,  or  rather  considering 
it  as  a  separate  and  independent  shunt  circuit  outside  of 
the  transformer,  as  can  approximately  be  done,  and  assum- 
ing the  primary  impedance  reduced  to  the  secondary  circuit 
as  equal  to  the  secondary  impedance, 


Substituting  this  in  the  equations  of  the  general  trans- 
former, we  get, 

=  -  nQe     1  +  -L  \r,  (r,  +  r)  +  sXl  (x,  +  *)] 


El  =  --  {  \f(ri+  r)  +  sax(xl  +  x) 
(n  +  + 


136.    If 

E  =  a  +  //?  =  E.M.F.,  in  complex  quantities, 
and 

/  =  c  -j-  j d  =  current  in  complex  quantities, 

the  power  is, 

/>  =    E,  I  |  =  .£  7  cos  (^,  7)  =  ac  +  y8^. 


202  AL  TERN  A  TING-CURRENT  PHENOMENA.      [.§  1 37 

Making  use  of  this,  and  denoting, 


*t* 


gives  : 

Secondary  output  of  the  transformer 


Internal  loss  in  secondary  circuit, 


** 
Total  secondary  power, 


Internal  loss  in  primary  circuit, 


\  . 

Total  electrical  output,  plus  loss, 

=  Pl  +  tf  +  PJ  =  ^Y  (r  +  2  ri)  =  f  w  (r  +  2  ^). 

V  V  7     . 

Total  electrical  input  of  primary, 

YV  +  >i  +  ^0  =  «/  (r  +  n  +  J^). 


Hence,  mechanical  output  of  transformer, 

P=  P0  -  Pl  =  w  (1  -  s)  (r+rj. 
Ratio, 

mechanical  output  P  _  1   —  S  _  speed 

total  secondary  power          p     ,\       v  I  ~  ~  ~    slip    *.  _       .     - 

137.    Thus, 

In  a  general  alternating  transformer  of  ratio  of  turns,  a, 
and  ratio  of  frequencies,  s,  neglecting  exciting  current,  it  is  : 

Electrical  input  in  primary, 

p  _    snfe^r+^  +  sr^ 


§138]       AL  TERN  A  TING-CURRENT  TRA  NSFORMER.  203 

Mechanical  output, 

p  =  s  (*  —  *] 


Electrical  output  of  secondary 

•V  o  O        O 

s2  n?  e2-  r 


p 
*i  - 


Losses  in  transformer, 

2sanl*e2.rl 


i  _i    pi        pi  _ 
t   -h  /i   =  ^   = 


Of  these  quantities,  P1  and  P1  are  always  positive  ;  PQ 
and  .P  can  be  positive  or  negative,  according  to  the  value 
of  s.  Thus  the  apparatus  can  either  produce  mechanical 
power,  acting  as  a  motor,  or  consume  mechanical  power; 
and  it  can  either  consume  electrical  power  or  produce 
electrical  power,  as  a  generator. 

138.    At 

j  =  0,  synchronism,  PQ  =  0,  P  =  0,  Pl  =  0. 
At     0  <  s  <  1,  between  synchronism  and  standstill. 

Pl ,  P  and  P0  are  positive  ;  that  is,  the  apparatus  con- 
sumes electrical  power  PQ  in  the  primary,  and  produces 
mechanical  power  P  and  electrical  power  Pl  +  P^  in  the 
secondary,  which  is  partly,  P^ ,  consumed  by  the  internal 
secondary  resistance,  partly,  P1 ,  available  at  the  secondary 
terminals. 

In  this  case  it  is  : 

PI  +  PS  _       s 
~P~      =l-s' 

that  is,  of  the  electrical  power  consumed  in  the  primary 
circuit,  PQ,  a  part  PQl  is  consumed  by  the  internal  pri- 
mary resistance,  the  remainder  transmitted  to  the  secon- 
dary, and  divides  between  electrical  power,  P1  -f  P^t  and 
mechanical  power,  P,  in  the  proportion  of  the  slip,  or  -drop 
below  synchronism,  s,  to  the  speed :  1  —  s. 


204  AL  TERN  A  TING-CURRENT  PHENOMENA  ,        [§138 

In  this  range,  the  apparatus  is  a  motor. 
At  s  >  1  ;  or,  backwards  driving, 

P  <  0,  or  negative  ;  that  is,  the  apparatus  requires  mechanical 
power  for  driving. 

It  is  then  :  PQ  -  P0l  -  J\l  <  P1  ; 

that  is  :  the  secondary  electrical  power  is  produced  partly 
by  the  primary  electrical  power,  partly  by  the  mechanical 
power,  and  the  apparatus  acts  simultaneously  as  trans- 
former and  as  alternating-current  generator,  with  the  sec- 
ondary as  armature. 

The  ratio  of  mechanical  input  to  electrical  input  is  the 
ratio  of  speed  to  synchronism. 

In  this  case,  the  secondary  frequency  is  higher  than  the 
primary. 

At  P  <  0,  beyond  synchronism, 

P  <  0  ;  that  is,  the  apparatus  has  to  be  driven  by  mechanical 

power. 
PQ  <  0  ;    that  is,  the  primary  circuit  produces  electrical  power 

from  the  mechanical  input. 

At  r'+ri+  Jfi  =  0,    or,    s  <  -  ^t^1  ; 

r 

the  electrical  power  produced  in  the  primary  becomes  less 
than  required  to  cover  the  losses  of  power,  and  PQ  becomes 
positive  again. 
We  have  thus  : 


consumes  mechanical  and  primary  electric  power  ;  produces 
secondary  electric  power. 

_  L+l!  <  ,  <  o 
r 

consumes   mechanical,    and    produces    electrical    power    in 
primary  and  in  secondary  circuit. 


§139]      ALTERNATING-CURRENT  TRANSFORMER. 


205 


consumes  primary  electric  power,  and  produces  mechanical 
and  secondary  electrical  power. 

Vf< 

consumes   mechanical^  and  primary  electrical  power ;  pro- 
duces secondary  electrical  power. 


GENERAL  ALTERNATE  CURRENT  TRANSFORMER 


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2   24 

Fig.   104. 

139.    As  an  instance,  in  Fig.  1  are  plotted,  with  the  slip 
s  as  abscissae,  the  values  of : 

Secondary  electrical  output  as  Curve      I. 
Total  internal  loss  as  Curve    II. 

Mechanical  output  as  Curve  III. 

Primary  electrical  input        as  Curve  IV. 

for  the  values  : 

n^e  =  100.0 ;  r     =         A : 


.3; 


k,    = 


9  • 

•^  5 


206  ALTERNATING-CURRENT  PHENOMENA.        [§139 

hence,  p        16,000  j*. 

' 


4,000^  +(5  +  j). 

1  +  ^2 
_  20,000  J  (1  -  J) 


14O]  INDUCTION  MOTOR.  207 


"CHAPTER  xv, 

INDUCTION  MOTOR. 

140.  A  specialization  of  the  general  alternating-current 
transformer  is  the  induction  motor.  It  differs  from  the  sta- 
tionary alternating-current  transformer  in  so  far  as  the  two 
sets  of  electric  circuits  —  the  primary  or  excited,  and  the 
secondary  or  induced,  circuits  —  are  movable  with  regard  to 
each  other  ;  and  that  in  general  a  number  of  primary  and 
a  number  of  secondary  circuits  are  used,  angularly  displaced 
around  the  periphery  of  the  motor,  and  containing  E.M.Fs. 
displaced  in  phase  by  the  same  angle.  This  multi-circuit 
arrangement  has  the  object  always  to  retain  secondary  cir- 
cuits in  inductive  relation  to  primary  circuits,  in  spite  of 
their  relative  motion. 

The  result  of  the  relative  motion  between  primary  and 
secondary  is,  that  the  E.M.Fs.  induced  in  the  secondary  or 
the  motor  armature  are  not  of  the  same  frequency  as  the 
E.M.F.  impressed  upon  the  primary,  but  of  a  frequency 
which  is  the  difference  between  the  impressed  frequency 
and  the  frequency  of  rotation,  or  equal  to  the  "  slip,"  that 
is,  the  difference  between  synchronism  and  speed  (in 
cycles). 

Hence,  if 

^V  =  frequency  of  main  or  primary  E.M.F., 

and  s  =  percentage  slip  ; 

sN~  =  frequency  of  armature  or  secondary  E.M.F., 

and  (1  —  s)  JV  =  frequency  of  rotation  of  armature. 

In  its  reaction  upon  the  primary  circuit,  however,  the 
armature  current  is  of  the  same  frequency  as  the  primary 


208  ALTERNATING-CURRENT  PHENOMENA,          [§141 

current,  since  it  is  carried  around  mechanically,  with  such 
a  frequency  as  always  to  have  the  same  phase  relation,  in 
the  same  position,  with  regard  to  the  primary  current. 

141.  Let  the  primary  system  consist  of/  equal  circuits, 
displaced  angularly  in  space  by  1  //  of  a  period,  that  is, 
1  //  of  the  width  of  two  poles,  and  excited  by  /  E.M.Fs. 
displaced  in  phase  by  1  j p  of  a  period ;  that  is,  in  other 
words,  let  the  field  circuits  consist  of  a  symmetrical  /-phase 
system. 

Analogously,  let  the  armature  or  secondary  circuits  con- 
sist of  a  symmetrical  /rphase  system. 

Let 

n   =  number  of  primary  turns  per  circuit  or  phase  ; 
TZj  =  number  of  secondary  turns  per  circuit  or  phase  ; 

a   =  — ^—  =  ratio  of  total  primary  turns  to  total  secondary  turns 
«iA 
or  ratio  of  transformation. 

Since  the  number  of  secondary  circuits  and  number  of 
turns  of  the  secondary  circuits,  in  the  induction  motor  — 
like  in  the  stationary  transformer  —  is  entirely  unessential, 
it  is  preferable  to  reduce  all  secondary  quantities  to  the 
primary  system,  by  the  ratio  of  transformation,  a  ;  thus 

if  EI   =  secondary  E.M.F.  per  circuit,  £l  =  a  £/ 

=  secondary  E.M.F.  per  circuit  reduced  to  primary 

system ; 

2 1 

if  Ii     =  secondary  current  per  circuit,  /!    =  — 

a 

=  secondary  current  per  circuit  reduced  to  primary 
system ; 

if  ri     =  secondary  resistance  per  circuit,  r±    =  a2  r± 

=  secondary  resistance  per  circuit  reduced  to  pri- 
mary system  ; 

if  Xi    —  secondary  reactance  per  circuit,  xl   =  a2  x^ 

=  secondary  reactance  per  circuit  reduced  to  pri- 
mary system ; 


§  142]  INDUCTION  MOTOR.  209 


if  Zi     =  secondary  impedance  per  circuit,          z± 

=  secondary  impedance  per  circuit  reduced  to  pri- 
mary system  ; 

'  '  '\  ''4  *     * 

that  is,  the  number  of  secondary  circuits  and  of  turns  per 

secondary  circuit  is'assumed  the  same  as  in  the  primary 
system. 

In  the  following  discussion,  as  secondary  quantities  ex- 
clusively, the  values  reduced  to  the  primary  system  shall 
be  used,  so  that,  to  derive  the  true  secondary  values,  these 
quantities  have  to  be  reduced  backwards  again  by  the  factor 

nfi 
—  — 


142.    Let 
4>  =  total  maximum  flux  of  the  magnetic  field  per  motor  pole. 

It  is  then 

E  =  V27r;*7v~<J>10-8  =  effective  E.M.F.  induced  by  the  mag- 
netic field  per  primary  circuit. 

Counting  the  time  from  the  moment  where  the  rising 
magnetic  flux  of  mutual  induction  <2>  (flux  interlinked  with 
both  electric  circuits,  primary  and  secondary)  passes 
through  zero,  in  complex  quantities,  the  magnetic  flux  is 

denoted  by 

*=/*, 

and  the  primary  induced  E.M.F., 

E=  ~e-, 
where 

e  =  V2  irnN$  10~8  may  be  considered  as  the  "  Active  E.M.F. 
of  the  motor." 

Since  the  secondary  frequency  is  s  N,  the  secondary 
induced  E.M.F.  (reduced  to  primary  system)  is 

El  =  —  se. 


210  AL  TERN  A  TING-CURRENT  PHENOMENA.        [  §  142 

Let 

I0  =  exciting  current,  or  current  passing  through  the  motor,  per 

primary  circuit,  when  doing  no  work  (at  synchronism), 
and 


primary  admittance  per  circuit  =  —  . 


It  is  thus 


ge  =  magnetic  energy  current,  ge*  =  loss  of  power  by  hysteresis 
(and  eddy  currents)  per  primary  coil. 

Hence 

pgei=  total  loss  of  energy  by  hysteresis  and  eddys,  as  calculated 
according  to  Chapter  X. 

be  =  magnetizing  current, 
and 

nb  e  =  effective  M.M.F.  per  primary  circuit  ; 

hence  , 

P.  nbe  =  total  effective  M.M.F.  ; 

and 

-±—  nbe  =  total  maximum  M.M.F.,  as  resultant  of  the  M.M.Fs.  of 
v^       the  /-phases,  combined  by  the  parallelogram  of  M.M.Fs.* 

If   (31  =  reluctance  of  magnetic  circuit  per  pole,  as  dis- 
cussed in  Chapter  X.,  it  is 


V2 

Thus,  from  the  hysteretic  loss,  and  the  reluctance,  the 
constants,  g  and  £,'and  thus  the  admittance,  Y.  are  derived. 
Let 

r    =  resistance  per  primary  circuit  ; 
z    =  reactance  per  primary  circuit  ; 

thus, 

Z  —  r  —  jx  =  impedance  per  primary  circuit  ; 

*  Complete  discussion  hereof,  see  Chapter  XXIII. 


§  143]  INDUCTION  MOTOR.  211 

/'!  =  resistance  per  secondary  circuit  reduced  to  primary  sys- 
tem ; 

xl  =  reactance  per  secondary  circuit  reduced  to  primary  system, 
at  full  frequency,  N;  • 

hence,  -  j^ 

sxi  =  reactance  per  secondary  circuit  at  slip  s\ 

and 

Zl    =  ri  —jsxl  =  secondary  internal  impedance. 

143.    It  is  now, 
Primary  induced  E.M.F., 

E    =  -e. 
Secondary  induced  E.M.F., 

EI  =  —  se. 
Hence, 
Secondary  current, 


^i  n  —  jsxi 

Component  of  primary  current,  corresponding  thereto, 
//  =  _  /.  =        ".      ; 


Primary  exciting  current, 

70    =eY= 
hence, 

Total  primary  current, 


E.M.F.  consumed  by  primary  impedance, 
£,=  ZI 


e(r  -joe)  \  S-£  +  (g+jb)  \  ; 

\r^-jsx  \ 


212  ALTERNATING-CURRENT  PHENOMENA.         [§143 

E.M.F.  required  to  overcome  the  primary  induced  E.M.F., 

-£   =  e; 
hence, 

Primary  terminal  voltage, 


We  get  thus,  in  an  induction  motor,  at  slip  s  and  active 
E.M.F.  e,  it  is 

Primary  terminal  voltage, 
£.  =  e\l  +  s  (r  -/*>  +  (r-j*) 

r\   ~  JSX\ 

Primary  current, 


or,  in  complex  expression, 
Primary  terminal  voltage, 


Primary  current, 


To  eliminate  e,  we  divide,  and  get, 

Primary  current,  at  slip  j,  and  impressed  E.M.F.,  E0  : 

T     •=  s  +  Z1Y  E   . 

ZT^  +  SZ  +  ZZ^Y  ' 

or, 


—  jx)  +  (r-  jx)  fa  -  jsx^  (g  +  jb) 


Neglecting,    in    the    denominator,    the    small    quantity 
^Y,  it  is 


§144] 


INDUCTION  MOTOR, 


213 


l  +  sZ 


(r—jx) 


r> 


or,  expanded, 


)  +  rx  V  +  jrt  (r^-  xb)  +  ^2^ 


rbj]  + 


/== 


Hence,  displacement  of  phase  between  current  and  E.M.F., 
tanoi  = 


(rr  +  JV) 

Neglecting  the  exciting  current,  Iot  altogether,  that  is, 
setting  Y  =  0,  it  is, 

I=sE  (ri  +  J 

'o 


tan 


s  (x  + 


sr 


144.    In  graphic    representation,   the   induction   motor 
diagram  appears  as  follows  :  — 


\ 


214  ALTERNATING-CURRENT  PHENOMENA. 

Denoting  the  magnetism  by  the  vertical  vector  OQ  in 
Fig.  105,  the  M.M.F.  in  ampere-turns  per  circuit  is  repre- 
sented by  vector  OF,  leading  the  magnetism  O®  by  the 
angle  of  hysteretic  advance  a.  The  E.M.F.  induced  in 
the  secondary  is  proportional  to  the  slip  s,  and  represented 
by  O£l  at  the  amplitude  of  180°.  Dividing  OE^  by  a  in 
the  proportion  of  r^-^sx^,  and  connecting  a  with  the 
middle  b  of  the  upper  arc  of  the  circle  OE^  ,  this  line  inter- 
sects the  lower  arc  of  the  circle  at  the  point  1^  r^  .  Thus, 
Ol^r^  is  the  E.M.F.  consumed  by  the  secondary  resistance, 
and  OI^x^  equal  and  parallel  to  E^I^r^  is  the  E.M.F.  con- 
sumed by  the  secondary  reactance.  The  angle,  El  OI^  i\ 
=  Wj  is  the  angle  of  secondary  lag. 

The  secondary  M.M.F.  OGl  is  in  the  direction  of  the 
vector  Ol^r^.  Completing  the  parallelogram  of  M.M.Fs. 
with  OF  as  diagonal  and  OG1  as  one  side,  gives  the  primary 
M.M.F.  OG  as  other  side.  The  primary  current  and  the 
E.M.F.  consumed  by  the  primary  resistance,  represented  by 
Olr,  is  in  line  with  OG,  the  E.M.F.  consumed  by  the  pri- 
mary reactance  90°  ahead  of  OG,  and  represented  by  OIx, 
and  their  resultant  OIz  is  the  E.M.F.  consumed  by  the 
primary  impedance.  The  E.M.F.  induced  in  the  primary 
circuit  is  OEly  and  the  E.M.F.  required  to  overcome  this 
counter  E.M.F.  is  OE  equal  and  opposite  to  OE1  .  Com- 
bining OE  with  OIz  gives  the  primary  terminal  voltage 
represented  by  vector  OE0,  and  the  angle  of  primary  lag, 


145.  Thus  far  the  diagram  is  essentially  the  same  as 
the  diagram  of  the  stationary  alternating-current  trans- 
former. 

Regarding  dependence  upon  the  slip  of  the  motor,  the 
locus  of  the  different  quantities  for  different  values  of 
the  slip  s  is  determined  thus  : 

It  is  £l  =  s£lf 

O  A  -r-  /!  r  =  E  -r-  /!  s  x 


§145] 


INDUCTION  MOTOR. 


215 


constant. 


X1 


That  is,  /]_  r±  lies  on  a  half-circle  with  —1 E^  as  diameter. 


Fig.  106. 

That  means  Gl  lies  on  a  half-circle  g^  in  Fig.  106  with 
OA  as  diameter.  In  consequence  hereof,  G0  lies  on  half- 
circle  g0  with  FB  equal  and  parallel  to  OA  as  diameter. 

Thus  Ir  lies  on  a  half-circle  with  Z>//"  as  diameter,  which 
circle  is  perspective  to  the  circle  FB,  and  Ix  lies  on  a  half- 
circle  with  IK  as  diameter,  and  Is  on  a  half-circle  with  LN 
as  diameter,  which  circle  is  derived  by  the  combination  of 
the  circles  Ir  and  Ix. 


216  AL  TERNA  TING-CURRENT  PHENOMENA.         [  §  146 

The  primary  terminal  voltage  E0  lies  thus  on  a  half- 
circle  eo  equal  to  the  half-circle  1  'z,  and  having  to  point 
E  the  same  relative  position  as  the  half-circle  Iz  has  to 
point  0. 

This  diagram  corresponds  to  constant  intensity  of  the 
maximum  magnetism,  O&.  If  the  primary  impressed  voltage 
E0  is  kept  constant,  the  circle  e  o  of  the  primary  impressed 
voltage  changes  to  an  arc  with  O  as  center,  and  all  the  cor- 
responding points  of  the  other  circles  have  to  be  reduced 
in  accordance  herewith,  thus  giving  as  locus  of  the  other 
quantities  curves  of  higher  order  which  most  conveniently 
are  constructed  point  for  point  by  reduction  from  the  circle 
of  the  locus  in  Fig.  106. 

Torque  and  Power. 

146.  The  torque  developed  per  pole  by  an  eiectnc 
motor  equals  the  product  of  effective  magnetism,  <£/V2, 
times  effective  armature  M.M.F.,  F/^/2,  times  the  sine  of 
the  angle  between  both, 


sn 


If  n^  =  number  of  turns,  71  =  current,  per  circuit,  with 
/-armature  circuits,  the  total  maximum  current  polarization, 
or  M.M.F.  of  the  armature,  is 


V2 
Hence  the  torque  per  pole, 


If   d  =  the   number   of   poles   of    the   motor,   the    total 
torque  of  the  motor  is, 


2  V2 


§148]  INDUCTION  MOTOR.  21T 

The  secondary  induced  E.M.F.,  Elt  lags  90°  behind  the 
inducing  magnetism,  hence  reaches  a  maximum  displaced  in 
space  by  90°  from  the  position. of  maximum  magnetization. 
Thus,  if  the  secondary  current,  7j,  lags  behind  its  E.M.F., 
Ev  by  angle,  o>j ,  the  space  displacement  between  armature 
current  and  field  magnetism  is 

f,  <£)  =  90°+  w, 
hence,  sin  (<j>  /:)  =  cos  oij 

It  is,  however, 

cos  «!  = r*  , 


thus,  9  = 

substituting  these  values  in  the  equation  of  the  torque,  it  is 

dp  s  ^  e*  108 


or,  in  practical  (C.G.S.)  units, 


_  dp  s  r± 


The   Torque  of  the  Induction  Motor. 

At  the  slip  s,  the  frequency  N,  and  the  number  of  poles 
d,  the  linear  speed  at  unit  radius  is 


hence  the  output  of  the  motor, 

P=  TS 

or,  substituted, 


218  A  L  TERN  A  TING-CURRENT  PHENOMENA.        [  §  147 

The  Power  of  the  Induction  Motor. 

147.  We  can  arrive  at  the  same  results  in  a  different 
way  : 

By  the  counter  E.M.F.  e  of  the  primary  circuit  with 
current  /  =  70  +  7X  the  power  is  consumed,  el  =  eI0  -+-  el±. 
The  power  eI0  is  that  consumed  by  the  primary  hysteresis 
and  eddys.  The  power  e  7X  disappears  in  the  primary  circuit 
by  being  transmitted  to  the  secondary  system. 

Thus  the  total  power  impressed  upon  the  secondary 
system,  per  circuit,  is 

PI  ='/i 

Of  this  power  a  part,  -fi^/j,  is  consumed  in  the  secondary 
circuit  by  resistance.  The  remainder, 

P'  =  Il(e-  E& 

disappears  as  electrical  power  altogether  ;  hence,  by  the  law 
of  conservation  of  energy,  must  reappear  as  some  other 
form  of  energy,  in  this  case  as  mechanical  power,  or  as  the 
output  of  the  motor  (included  mechanical  and  secondary 
magnetic  friction). 

Thus  the  mechanical  output  per  motor  circuit  is 

P'  =  /!(<?-  £,). 
Substituting, 


rl—jsxl 
it  is 


hence,  since  the  imaginary  part  has  no  meaning  as  power, 
p,  =  r^sQ  —  s)  . 

*-  2  J_    c  2  v  2      ' 

r±  .  -\-  s  &i 


§148]  INDUCTION  MOTOR.  219 

.and  the  total  power  of  the  motor, 

At  the  linear  speed, 

*"  ±TtN    ,. 


at  unit  radius  the  torque  is 


In  the  foregoing,  we  found 

E0  = 
or,  approximately, 


.expanded,          ,     = 


•or,  eliminating  imaginary  quantities  : 

e     =  1 


Substituting  this  value  in  the  equations  of  torque  and  of 
power,  it  is, 


Maximum  Torque. 

148.    The  torque  of  the  induction  motor  is  a  maximum 
for  that  value  of  slip  j,  where 


220  ALTERNATING-CURRENT  PHENOMENA.         [§148 

dp  i\  £02  K 

~ 

d 
°r' 


expanded,  this  gives, 

r-~  +  *"*  -f-  (*!  -f  x)*  =  0, 

*i 


or, 


Substituting  this  in  the  equation  of  torque,  we  get  the 
value  of  maximum  torque, 


That  is,  independent  of  the  secondary  resistance,  r^  . 
The  power  corresponding  hereto  is,  by  substitution  of  st 
in  P,  _ 

__  P  Eo2  {  V'-2  +  (*i  +  ^)2  -  r,} 


This  power  is  not  the  maximum  output  of  the  motor, 
but  already  below  the  maximum  output.  The  maximum 
output  is  found  at  a  lesser  slip,  or  higher  speed,  while  at 
the  maximum  torque  point  the  output  is  already  on  the 
decrease,  due  to  the  decrease  of  speed. 

With  increasing  slip,  or  decreasing  speed,  the  torque  of 
the  induction  motor  increases  ;  or  inversely,  with  increasing 
load,  the  speed  of  the  motor  decreases,  and  thereby  the 
torque  increases,  so  as  to  carry  the  load  down  to  the  slip  st  ,. 
corresponding  to  the  maximum  torque.  At  this  point  of 
load  and  slip  the  torque  begins  to  decrease  again  ;  that  is, 
as  soon  as  with  increasing  load,  and  thus  increasing  slip, 
the  motor  passes  the  maximum  torque  point  ^  ,  it  "falls  out 
of  step,"  and  comes  to  a  standstill. 

Inversely,  the  torque  of  the  motor,  when  starting  from 
rest,  will  increase  with  increasing  speed,  until  the  maximum 


§  149  ]  IND  UC  TION  MO  TOR.  221 

torque  point  is  reached.  From  there  towards  synchronism 
the  torque  decreases  again. 

In  consequence  hereof,  the  part  of  the  torque-speed 
curve  below  the  maximum  torque  point  is  in  general 
unstable,  and  can  be^  observed  only  by  loading  the  motor 
with  an  apparatus,  whose  countertorque  increases  with  the 
speed  faster  than  the  torque  of  the  induction  motor. 

In  general,  the  maximum  torque  point,  st,  is  between 
synchronism  and  standstill,  rather  nearer  to  synchronism. 
Only  in  motors  of  very  large  armature  resistance,  that  is 
low  efficiency,  st  >  1,  that  is,  the  maximum  torque  falls 
below  standstill,  and  the  torque  constantly  increases  from 
synchronism  down  to  standstill. 

It  is  evident  that  the  position  of  the  maximum  torque 
point,  st)  can  be  varied  by  varying  the  resistance  of  the 
secondary  circuit,  or  the  motor  armature.  Since  the  slip 
at  the  maximum  torque  point,  st ,  is  directly  proportional  to 
the  armature  resistance,  r\ ,  it  follows  that  very  constant 
speed  and  high  efficiency  will  bring  the  maximum  torque 
point  near  synchronism,  and  give  small  starting  torque, 
while  good  starting  torque  means  a  maximum  torque  point 
at  low  speed ;  that  is,  a  motor  with  poor  speed  regulation 
and  low  efficiency. 

Thus,  to  combine  high  efficiency  and  close  speed  regu- 
lation with  large  starting  torque,  the  armature  resistance 
has  to  be  varied  during  the  operation  of  the  motor,  and  the 
motor  started  with  high  armature  resistance,  and  with  in- 
creasing speed  this  armature  resistance  cut  out  as  far  as 
possible. 

149.    If  st  =  I, 


it  is  r±  =      r2  +  (xl  +  x)*. 

In  this  case  the  motor  starts  with  maximum  torque,  and 
when  overloaded  does  not  drop  out  of  step,  but  gradually 
slows  down  more  and  more,  until  it  comes  to  rest. 


222  ALTERNATING-CURRENT  PHENOMENA.          [§15O 

If,  st  >  1, 


it  is  T-i  >  Vr2  +  (Xl  +  *)2. 

In  this  case,  the  maximum  torque  point  is  reached  only 
by  driving  the  motor  backwards,  as  countertorque. 
r  As  seen  above,  the  maximum  torque,  rt,  is  entirely  inde- 
pendent of  the  armature  resistance,  and  the  same  is  the 
current  corresponding  thereto,  independent  of  the  armature 
resistance.  Only  the  speed  of  the  motor  depends  upon  the 
armature  resistance. 

Hence  the  insertion  of  resistance  into  the  motor  arma- 
ture does  not  change  the  maximum  torque,  and  the  current 
corresponding  thereto,  but  merely  lowers  the  speed  at 
which  the  maximum  torque  is  reached. 

The  effect  of  resistance  inserted  into  the  induction 
motor  is  merely  to  consume  the  E.M.F.,  which  otherwise 
would  find  its  mechanical  equivalent  in  an  increased  speed, 
analogous  as  resistance  in  the  armature  circuit  of  a  continu- 
ous-current shunt  motor. 

Further  discussion  on  the  effect  of  armature  resistance 
is  found  under  "Starting  Torque." 

Maximum  Power. 

150.    The  power  of  an  induction  motor  is  a  maximum 

for  that  slip,  s  ,  where        ,  _ 

££  =  0; 
ds 


or,  since 


7>=  — 

£j(s 
rf/| 

expanded,  this  gives 

J*  ~  n  +  Vfa  +  r)2  +  (^  +  ^] 


§15O]  INDUCTION  MOTOR.  223 

substituted  in  P,  we  get  the  maximum  power, 

PE? 


This  result  has  a  simple  physical  meaning  :  (r±  -+-  r)  =  R 
is  the  total  resistance  of  the  motor,  primary  plus  secondary 
(the  latter  reduced  to  the  primary).  (x^  -f  x)  is  the  total 
reactance,  and  thus  V(rL  +  rf  +  (xl  +  ^r)2  =  Z  is  the 
total  impedance  of  the  motor.  Hence  it  is 


the  maximum  output  of  the  induction  motor,  at  the  slip, 


The  same  value  has  been  derived  in-  Chapter  IX.,  as  the 
maximum  power  which  can  be  transmitted  into  a  non- 
inductive  receiver  circuit  over  a  line  of  resistance  R,  and 
impedance  Z,  or  as  the  maximum  output  of  a  generator,  or 
of  a  stationary  transformer.  Hence  : 

The  maximum  output  of  an  induction  motor  is  expressed 
.by  the  same  formula  as  the  maximum  output  of  a  generator, 
or  of  a  stationary  transformer,  or  the  maximum  output  which 
tan  be  transmitted  over  an  inductive  line  into  a  non-inductive 
receiver  circuit. 

The  torque  corresponding  to  the  maximum  output  Pp  is, 


This   is   not   the   maximum   torque,    but   the    maximum 
torque,  rt  ,  takes  place  at  a  lower  speed,  that  is,  greater  slip, 


that  is,  st> 


224  AL  TERN  A  TING-CURRENT  PHENOMENA.          [  §  151 

It  is  obvious  from  these  equations,  that,  to  reach  as  large 
an  output  as  possible,  R  and  Z  should  be  as  small  as  possi- 
ble ;  that  is,  the  resistances  ^  +  r,  and  the  impedances,  Z, 
and  thus  the  reactances,  x±  -f  x,  should  be  small.  Since 
r\  -+-  r  is  usually  small  compared  with  xl  +  x,  it  follows,  that 
the  problem  of  induction  motor  design  consists  in  con- 
structing the  motor  so  as  to  give  the  minimum  possible 
reactances,  xl  -f-  x. 

Starting  Torque. 

151.    In  the  moment   of   starting  an   induction  motor, 

*  is  s  =  1  ; 

hence,  starting  current  : 


(ri  -y*i)  +  (r  -joe}  +  (ri  -/*i)  (r 

or,  expanded,  with  the  neglection  of  the  last  term  in  the 
denominator,  as  insignificant  : 


r 


and,  displacement  of  phase,  or  angle  of  lag, 
tan<3  =  (-^i  + 


[rx  +  r]  +  ^  [^  +  ^])  +  £  (r*!  -  flFf,) 

Neglecting  the  exciting  current,  g  =  0  =  b,  these  equa- 
tions assume  the  form  : 


r_ 


or,  eliminating  imaginary  quantities, 


and 


§  152]  INDUCTION  MOTOR.  225 

That  means,  that  in  starting  the  induction  motor  without 
additional  resistance  in  the  armature  circuit,  —  in  which  case 
^  -f-  x  is  large  compared  with  rv  -f  r,  and  the  total  impe- 
dance, Z,  small,  —  the  motor  takes  excessive  and  greatly 
lagging  curients. 

The  starting  torque  is 


=  dp  £0*    r^ 
~    ItrN     Z2' 

That  is,  the  starting  torque  is  proportional  to  the 
armature  resistance,  and  inverse  proportional  to  the  square 
of  the  total  impedance  of  the  motor. 

It  is  obvious  thus,  that,  to  secure  large  starting  torque, 
the  impedance  should  be  as  small,  and  the  armature  resis- 
tance as  large,  as  possible.  The  former  condition  is  the 
condition  of  large  maximum  output  and  good  efficiency 
and  speed  regulation  ;  the  latter  condition,  however,  means 
inefficiency  and  poor  regulation,  and  thus  cannot  properly 
be  fulfilled  by  the  internal  resistance  of  the  motor,  but  only 
by  an  additional  resistance  which  is  short-circuited  while 
the  motor  is  in  operation. 

152.    Since,  necessarily, 


and  since  the  starting  current  is,  approximately, 


r      _o 

~~  z' 


it  is, 


4  IT  N 


oo  —    - 

47T 


226  ALTERNATING-CURRENT  PHENOMENA.         [§  152 

•Would  be  the  theoretical  torque  developed  at  100  per  cent 
.efficiency,  and  power  factor  by  E.M.F.,  E0,  and  current,  /> 
at  synchronous  speed. 

It  is  thus,  TO  <  TOO, 

and  the  ratio  between  the  starting  torque,  TO,  and  the  theo- 
retical maximum  torque,  TOO,  gives  a  means  to  judge  the 
perfection  of  a  motor  regarding  its  starting  torque. 

This  ratio,  TO  j  roo  ,  reaches  .8  to  .9  in  the  best  motors. 

Substituting  I=E/Z  in  the  equation  of  starting 
torque,  it  assumes  the  form, 

.    dp 
T  —  -  — 


Since  4  TT  N I  d  =  synchronous  speed,  it  is  : 

The  starting  torque  of  the  induction  motor  is  equal  to  the 
resistance  loss  in  the  motor  armature,  divided  by  the  synchro- 
nous speed. 

The  armature  resistance  which  gives  maximum  starting 
torque  is 

77° =  •':','. 

dp  Ef  r, 

or,smce         ra  =  ^-^    (r+ry+(Xl+xy 


dr 
expanded,  this  gives, 


the  same  value  as  derived  in  the  paragraph  on  "  maximum 
torque." 

Thus,  adding  to  the  internal  armature  resistance  i\    in 
starting  the  additional  resistance, 


makes  the  motor  start  with  maximum  torque,  while  with  in- 
creasing speed  the  torque  constantly  decreases,  and  reaches 


§  153]  INDUCTION  MOTOR.  227 

zero  at  synchronism.  Under  these  conditions,  the  induc- 
tion motor  behaves  similar  to  the  continuous-current  series 
motor,  varying  in  the  speed,  with  the  load,  the  difference 
being,  however,  that  the  *  induction  motor  approaches  a 
definite  speed  at  no-load,  wmle  with  the  series  motor  the 
speed  indefinitely  increases  with  decreasing  load. 

153.  The  additional  armature  resistance,  /Y',  required 
to  give  a  certain  starting  torque,  is  found  from  the  equation 
of  starting  torque  : 

Denoting  the  internal  armature  resistance  by  r±,  total 
armature  resistance  is  r^  =  r-l  +  1\'  , 

and  thus,  _  d  p  E*  r{  -\-  r"  _ 

~~~ 
hence, 


This  gives  two  values,  one  above,  the  other  below,  the 
maximum  torque  point. 

Choosing  the  positive  sign  of  the  root,  we  get  a  larger 
armature  resistance,  a  small  current  in  starting,  but  the 
torque  constantly  decreases  with  the  speed. 

Choosing  the  negative  sign,  we  get  a  smaller  resistance, 
a  large  starting  current,  and  with  increasing  speed  the 
torque  first  increases,  reaches  a  maximum,  and  then  de- 
creases again  towards  synchronism. 

These  two  points  correspond  to  the  two  points  of  the 
speed-torque  curve  of  the  induction  motor,  in  Fig.  107, 
giving  the  desired  torque  T0. 

The  smaller  value  of  r±  will  give  fairly  good  speed  reg- 
ulation, and  thus  in  small  motors,  where  the  comparatively 
large  starting  current  is  no  objection,  the  permanent  arma- 
ture resistance  may  be  chosen  to  represent  this  value. 

The  larger  value  of  r"  allows  to  start  with  minimum 
current,  but  requires  cutting  out  of  the  resistance  after  the 
start,  to  secure  speed  regulation  and  efficiency. 


228      ALTERNATING-CURRENT  PHENOMENA.     [§§154,155 

Synchronism. 
154.    At  synchronism,  s  =  0,  it  is  : 

f.  = 

or,  rationalized  : 


that  is,  power  and  torque  are  zero.  Hence,  the  induction 
motor  can  never  reach  complete  synchronism,  but  must 
slip  sufficiently  to  give  the  torque  consumed  by  friction. 

Running  near  Synchronism. 

155.  When  running  near  synchronism,  at  a  slip  s  above 
the  maximum  output  point,  where  s  is  small,  from  .02  to 
.05  at  full  load,  the  equations  can  be  simplified  by  neglect- 
ing terms  with  s,  as  of  higher  order. 

It  is,  current  : 

r       *+r*(g+jl>)  F  . 

1    =  -     JL0  , 

r\ 

or,  eliminating  imaginary  quantities  : 


j-  i  i     S 


r\ 
angle  of  lag  :  « xl  + 


-}-  rfg  s  + 


or,  inversely : 

s  - 
~ 


§§156,157]  INDUCTION  MOTOR.  229 

that  is  : 

Near  synchronism,  the  slip,  s,  of  an  induction  motor,  or 
its  drop  in  speed,  is  proportional  to  the  armature  resistance 
i\  and  to  the  power,  P,  or  torque. 


Induction   Generator. 

156.  In  the  foregoing,  the  range  of  speed  from  s  =  1, 
standstill,  to  s  =  0,  synchronism,  has  been  discussed.      In 
this  range  the  motor  does  mechanical  work. 

It  consumes  mechanical  power,  that  is,  acts  as  generator 
•or  as  brake,  outside  of  this  range. 

For,  s  >  1,  backwards  driving,  P  becomes  negative, 
representing  consumption  of  power,  while  s  remains  posi- 
tive ;  hence,  since  the  direction  of  rotation  has  changed, 
represents  consumption  of  power  also.  All  this  power  is 
consumed  in  the  motor,  which  thus  acts  as  brake. 

For,  s  <  0,  or  negative,  P  and  r  become  negative,  and 
the  machine  becomes  an  electric  generator,  converting  me- 
chanical into  electric  energy. 

Substituting  in  this  case  :  sl  =  —  s,  where  k±  is  the 
acceleration,  or  the  slip  of  the  machine  above  synchronism, 
we  derive  the  equations  of  the  induction  generator,  which 
are  the  same  as  those  of  the  induction  motor,  except  that 
the  sign  before  the  "  slip  "  is  reversed. 

Again  a  maximum  torque  point,  and  a  maximum  output 
point  are  found,  and  the  torque  and  power  increase  from 
zero  at  synchronism  up  to  a  maximum  point,  and  then  de- 
crease again,  while  the  current  constantly  increases. 

157.  The  induction  generator  differs  from  the  standard 
alternating-current  generator  essentially,  in  so  far  as  it  has 
no  definite  frequency  of  its  own,   but  can  operate  at   any 
frequency  above  that  corresponding   to  its  speed.      But  it 
can  generate  electric  energy  only  when  in  circuit  with  an 
alternating-current  apparatus  of    definite  frequency,  as  an 


230 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


[§158 


alternator   or   synchronous   motor.      That   is,  the  induction, 
generator  requires-  a  "frequency  setter"  for  its  operation. 

When  operating  in  parallel  with  standard  alternators, 
the  phase  relation  of  the  current  issuing  from  the  induction 
generator  mainly  depends  —  besides  upon  the  slip  —  upon 
the  self-induction  of  the  induction  generator,  and  can  be 
varied  thereby. 

v "  Hence  the  induction  generator  can  be  used  to  control 
the  phase  relation  in  an  alternating-current  circuit. 

When  connected  in  series  in  a  circuit,  the  E.M.F.  of  the 
induction  generator  is  approximately  proportional  to  the 
current.  Thus  it  can  be  used  as  booster,  to  add  voltage 
to  a  line  in  proportion  to  the  current  passing  therein. 


Example. 

158.    As  an   instance  are   shown,   in  Fig.  107,   charac- 
teristic curves  of   a   20  horse-power  three-phase  induction 


Amperes 
140    160    ISO    200  220    240   260    280    300   320    310 


Fig.  107.    Speed  Characteristics  of  Induction  Motor. 


§158] 


INDUCTION  MOTOR. 


231 


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Fig.  108.     Current  Characteristics  of  Induction  Motor. 

motor,  of  900  revolutions  synchronous  speed,  8  poles,  fre- 
quency of  60  cycles. 

The  impressed  E.M.F.  is  110  volts  between  lines,  and 
the  motor  star-connected,  hence  the  E.M.F.  impressed  per 
circuit  :  n() 

-  =  60.0  :    or  E0  =  63.5. 


The  constants  of  the  motor  are  : 

Primary  admittance,  Y  =  .1  -|"  .4  /. 
Primary  impedance,  Z  =  .03  —  .09  /. 
Secondary  impedance,  Z^  =  .02  —  .085  /. 


232 


AL  TERN  A  TING-CURRENT  PHENOMENA.         [  §  158 


In  Fig.  107  is  shown,  with  the  speed  in  per  cent  of 
synchronism,  as  abscissae,  the  torque  in  kilogrammetres, 
as  ordinates,  in  drawn  lines,  for  the  values  of  armature 
resistance : 

r^  —  .02    :  short  circuit  of  armature,  full  speed. 

r^  =  .045  :  .025  ohms  additional  resistance. 

r±  =  .16    :  .16   ohms    additional    resistance,   maximum    starting 

torque. 
TI  =  .75    :  .73  ohms  additional  resistance,  same  starting  torque 

as       =  .045. 


Fig.  109.    Speed  Characteristics  of  Induction  Motor. 

On  the  same  Figure  is  shown  the  current  per  line,  in 
dotted  lines,  with  the  verticals  or  torque  as  abscissae,  and 
the  horizontals  or  amperes  as  ordinates.  To  the  same 
torque  always  corresponds  the  same  current,  no  matter 
what  the  speed  be. 

On  Fig.  108  is  shown,  with  the  current  input  per  line 
as  abscissae,  the  torque  in  kilogrammetres  and  the  output 


§158]  .INDUCTION  MOTOR.  23S 

in  kilowatts  as  ordinates  in  drawn  lines,  and  the  speed  and 
the  magnetism,  in  per  cent  of  their  synchronous  values, 
as  ordinates  in  dotted  lines,  for  the  armature  resistance 
r±  =  .02,  or  short  circuit.  *  .<| 

In  Fig.  109  is  shown,  wifcii  the  speed,  in  per  cent  of 
synchronism,  as  abscissae,  the  torque 'in  drawn  line,  and 
the  output  in  dotted  line,  for  the  value  of  armature  resis- 
tance ;-!  =  .045,  for  the  whole  range  of  speed  from  120  per 
cent  backwards  speed  to  220  per  cent  beyond  synchronism, 
showing  the  two  maxima,  the  motor  maximum  at  s  =  .25, 
and  the  generator  maximum  at  s  =  —  25. 


234  ALTERNATING-CURRENT  PHENOMENA.         [§159 


CHAPTER    XVI. 

ALTERNATING-CURRENT    GENERATOR. 

159.  In  the  alternating-current  generator,  E.M.F.  is 
induced  in  the  armature  conductors  by  their  relative  motion 
through  a  constant  or  approximately  constant  magnetic 
field. 

When  yielding  current,  two  distinctly  different  M.M.Fs. 
are  acting  upon  the  alternator  armature  —  the  M.M.F.  of 
the  field  due  to  the  field-exciting  spools,  and  the  M.M.F. 
of  the  armature  current.  The  former  is  constant,  or  approx- 
imately so,  while  the  latter  is  alternating,  and  in  synchro- 
nous motion  relatively  to  the  former  ;  hence,  fixed  in  space 
relative  to  the  field  M.M.F.,  or  uni-directional,  but  pulsating 
in  a  single-phase  alternator.  In  the  polyphase  alternator, 
when  evenly  loaded  or  balanced,  the  resultant  M.M.F.  of 
the  armature  current  is  more  or  less  constant. 

The  E.M.F.  induced  in  the  armature  is  due  to  the  mas:- 

o 

netic  flux  passing  through  and  interlinked  with  the  arma- 
ture conductors.  This  flux  is  produced  by  the  resultant  of 
both  M.M.Fs.,  that  of  the  field,  and  that  of  the  armature. 

On  open  circuit,  the  M.M.F.  of  the  armature  is  zero,  and 
the  E.M.F.  of  the  armature  is  due  to  the  M.M.F.  of  the 
field  coils  only.  In  this  case  the  E.M.F.  is,  in  general,  a 
maximum  at  the  moment  when  the  armature-  coil  faces  the 
position  midway  between  adjacent  field  coils,  as  shown  in 
Fig.  110,  and  thus  incloses  no  magnetism.  The  E.M.F. 
wave  in  this  case  is,  in  general,  symmetrical. 

An  exception  from  this  statement  may  take  place  only 
in  those  types  of  alternators  where  the  magnetic  reluctance 
of  the  armature  is  different  in  different  directions  ;  thereby, 


§160]  ALTERNATING-CURRENT  GENERATOR.  235 

during  the  synchronous  rotation  of  the  armature,  a  pulsa- 
tion of  the  magnetic  flux  passing  through  it  is  produced. 
This  pulsation  of  the  magnetic  flux  induces  E.M.F.  in  the 
field  spools,  and  thereby  makes  the  field  current  pulsating 
also.  Thus,  we  have,  in  this  ga.se,  even  on  open  circuit,  no 


Fig.  770. 

rotation  through  a  constant  magnetic  field,  but  rotation 
through  a  pulsating  field,  which  makes  the  E.M.F.  wave 
unsymmetrical,  and  shifts  the  maximum  point  from  its  the- 
oretical position  midway  between  the  field  poles.  In  gen- 
eral this  secondary  reaction  can  be  neglected,  and  the  field 
M.M.F.  be  assumed  as  constant. 

160.  The  relative  position  of  the  armature  M.M.F.  with 
respect  to  the  field  M.M.F.  depends  upon  the  phase  rela- 
tion existing  in  the  electric  circuit.  Thus,  if  there  is  no 
displacement  of  phase  between  current  and  E.M.F.,  the 
current  reaches  its  maximum  at  the  same  moment  as  the 
E.M.F. ;  or,  in  the  position  of  the  armature  shown  in  Fig. 
110,  midway  between  the  field  poles.  In  this  case  the  arma- 
ture current  tends  neither  to  magnetize  nor  demagnetize  the 
field,  but  merely  distorts  it  ;  that  is,  demagnetizes  the  trail- 
ing-pole  corner,  a,  and  magnetizes  the  leading-pole  corner, 
.b.  A  change  of  the  total  flux,  and  thereby  of  the  resultant 
E.M.F.,  will  take  place  in  this  case  only  when  the  magnetic 
densities  are  so  near  to  saturation  that  the  rise  of  density 
at  the  leading-pole  corner  will  be  less  than  the  decrease  of 


236 


A  L  TERN  A  TING-CURRENT  PHENOMENA .         [  §  160 


density  at  the  trailing-pole  corner.  Since  the  internal  self- 
inductance  of  the  alternator  alone  causes  a  certain  lag  of 
the  current  behind  the  induced  E.M.F.,  this  condition  of  no 
displacement  can  exist  only  in  a  circuit  with  external  nega- 
tive reactance,  as  capacity,  etc. 

If  the  armature  current  lags,  it  reaches  the  maximum 
later  than  the  E.M.F.  ;  that  is,  in  a  position  where  the 
armature  coil  partly  faces  the  following-field  pole,  as  shown 
in  diagram  in  Fig.  111.  Since  the  armature  current  flows 


Fig.  777. 


in  opposite  direction  to  the  current  in  the  following-field 
pole  (in  a  generator),  the  armature  in  this  case  will  tend  to 
demagnetize  the  field. 

If,  however,  the  armature  current  leads, — that  is,  reaches 
its  maximum  while  the  armature  coil  still  partly  faces  the 


Fig.  112. 


preceding-field  pole,  as  shown  in  diagram  Fig.  112, — it  tends 
to  magnetize  this  field  coil,  since  the  armature  current  flows 
in  the  same  direction  with  the  exciting  current  of  the  pre- 
ceding-field spools. 


§161]  ALTERNATING-CURRENT  GENERATOR.  237 

Thus,  with  a  leading  current,  the  armature  reaction  of 
the  alternator  strengthens  the  field,  and  thereby,  at  con- 
stant-field excitation,  increases  the  voltage  ;  with  lagging 
current  it  weakens  the  field,  and  thereby  decreases  the  vol- 
tage in  a  generator.^  Obviously,  the  opposite  holds  for  a 
synchronous  motor,  in  which  the  direction  of  rotation  is 
opposite  ;  and  thus  a  lagging  current  tends  to  magnetize, 
a  leading  current  to  demagnetize,  the  field. 

161.  The  E.M.F.  induced  in  the  armature  by  the  re- 
sultant magnetic  flux,  produced  by  the  resultant  M.M.F.  of 
the  field  and  of  the  armature,  is  not  the  terminal  voltage 
of  the  machine;  the  terminal  voltage  is  the  resultant  of  this 
induced  E.M.F.  and  the  E.M.F.  of  self-inductance  and  the 
E.M.F.  representing  the  energy  loss  by  resistance  in  the 
alternator  armature.  That  is,  in  other  words,  the  armature 
current  not  only  opposes  or  assists  the  field  M.M.F.  in  cre- 
ating the  resultant  magnetic  flux,  but  sends  a  second  mag- 
netic flux  in  a  local  circuit  through  the  armature,  which 
flux  does  not  pass  through  the  field  spools,  and  is  called  the. 
magnetic  flux  of  armature  self-inductance. 

•  Thus  we  have  to  distinguish  in  an  alternator  between 
armature  reaction,  or  the  magnetizing  action  of  the  arma- 
ture upon  the  field,  and  armature  self-inductance,  or  the 
E.M.F.  induced  in  the  armature  conductors  by  the  current 
flowing  therein.  This  E.M.F.  of  self-inductance  is  (if  the 
magnetic  reluctance,  and  consequently  the  reactance,  of 
the  armature  circuit  is  assumed  as  constant)  in  quadrature 
behind  the  armature  current,  and  will  thus  combine  with 
the  induced  E.M.F.  in  the  proper  phase  relation.  This 
means  that,  if  the  armature  current  lags,  the  E.M.F.  of 
self-inductance  will  be  more  than  90°  behind  the  induced 
E.M.F.,  and  therefore  in  partial  opposition,  and  will  reduce 
the  terminal  voltage.  On  the  other  hand,  if  the  armature 
current  leads,  the  E.M.F.  of  self-inductance  will  be  less 
than  90°  behind  the  induced  E.M.F.,  or  in  partial  conjunc- 


238     ALTERNATING-CURRENT  PHENOMENA.     [§§162,163 

tion  therewith,  and  increase  the  terminal  voltage.  This 
means  that  the  E.M.F.  of  self-inductance  increases  the  ter- 
minal voltage  with  a  leading,  and  decreases  it  with  a  lagging 
current,  or,  in  other  words,  acts  in  the  same  manner  as  the 
armature  reaction. 

For  this  reason  both  actions  can  be  combined  in  one, 
and  represented  by  what  is  called  the  synchronous  reactance 
of  the  alternator. 

In  the  following,  we  shall  represent  the  total  reaction  of 
the  armature  of  the  alternator  by  the  one  term,  synchronous 
reactance.  While  this  is  not  exact,  as  stated  above,  since 
the  reactance  should  be  resolved  into  the  magnetic  reaction 
due  to  the  magnetizing  action  of  the  armature  current, 
and  the  electric  reaction  due  to  the  self-induction  of  the 
armature  current,  it  is  in  general  sufficiently  near  for  prac- 
tical purposes,  and  well  suited  to  explain  the  phenomena 
taking  place  in  the  alternator  under  the  various  conditions 
of  load. 

162.  This  synchronous  reactance,  x,  is  frequently  not 
constant,  but  is  pulsating,  owing  to  the  synchronously  vary- 
ing reluctance  of    the  armature  magnetic  circuit,  and  tke 
field  magnetic  circuit ;   it  may,  however,  be  considered  in 
what   follows   as   constant;    that  is,    the    E.M.Fs.    induced 
thereby  may  be  represented  by  their  equivalent  sine  waves. 
A  specific  discussion  of  the  distortions  of  the  wave  shape 
due  to  the  pulsation  of  the  synchronous  reactance  is  found 
in  Chapter  XX.      The  synchronous  reactance,  x,  is  not  a 
true  reactance  in  the  ordinary  sense  of  the  word,  but  an 
equivalent  or  effective  reactance. 

163.  Let  E0  =  induced  E.M.F.  of  the  alternator,  or  the 
E.M.F.   induced   in    the    armature    coils    by  their   rotation 
through  the  constant  magnetic  field  produced  by  the  cur- 
rent in  the  field  spools,  or  the  open  circuit  voltage  of  the 
alternator. 


§164]  ALTERNATING-CURRENT  GENERATOR.  239 

Then  E0  =  V2  TT  «  .V  J/  10  ~  8  ; 

where 

n    =  total  number  of  turn's  in  series  on  the  armature, 

N  =  frequency,  ^ 

M  =  total  magnetic  flux  per  field  pole. 

Let  oc0  =  synchronous  reactance, 

r0  =  internal  resistance  of  alternatof  ; 
then  Z0  =  r0  —  j  x0  =  internal  impedance. 

If  the  circuit  of  the  alternator  is  closed  by  the  external 
impedance, 


and  current 


or, 


E0 


and,  terminal  voltage, 

E  =  IZ  =  E0  -  IZ0 
E0(r  —  jx) 


(r0  +  r)  —  j  (x0  +  x)  ' 
or, 


H-  (x0 

1 


i/1  +  2  r°r2+  ^Lf  4-  5£±J| 

or,  expanded  in  a  series, 

„        ~  (,        r0r+  *0.y     2  (r0r  +  x0x)  —  (r0x  +  x0r) ^        \ 

0  j  J  r2    i    ^2    +  7  2  (r2  +  jc2)  "  j  * 


As  shown,  the  terminal  voltage  varies  with  the  condi- 
tions of  the  external  circuit. 

164.    As   an   instance,   in   Figs.   113-118,   at    constant 

induced,  the  E.M.F., 

E0  =  2500 ; 


240 


AL  TERN  A  TING-CURRENT  PHENOMENA.         [§  164 


and  the  values  of  the  internal  impedance, 
Z0  =  r0  -jx0  =  1  -  lOy. 

With  the  current  /  as  abscissae,  the  terminal  voltages  E 
as  ordinates  in  drawn  line,  and  the  kilowatts  output,  =  72  r, 
in  dotted  lines,  the  kilovolt-amperes  output,  =  IE,  in  dash- 


• 

^  ^ 

, 

1* 

S\ 
\ 

—  •  — 

-•^ 

•^^.^^ 

j 

' 

\ 
\ 

N. 

^^""•l 

*^ 

^ 

/ 

\ 
1 

\ 

/ 

\ 

N 

\ 

^£ 

\ 

\ 
\ 

/ 

—S 

9 

\ 

\ 
\ 

4 

\ 

t 
\ 

\ 

45* 

Jj 

\ 

I 

>  II 

II    0 
O-O 

'/ 

\ 

o  o 

X     X 

V 

\ 

\ 
\ 

/ 

F 

ELD 

CHA 

•1AC1 

ERIS 

TIC 

\ 

\ 

I 

1 

°l 

250( 

R  = 

3f  z  = 

E.   x 

MOj, 

=  0 

J 
\\ 

/ 

\ 

j 

Arr 

P3. 

\ 

0       .40         60         80        100       120       140       160       180       200       20       240       2 

0 

Fig.  113.    Field  Characteristic  of  Alternator  on  Non-inductive  Load. 

dotted  lines,  we  have,  for  the  following  conditions  of  external 
circuit : 

In  Fig.  113,  non-inductive  external  circuit,  x  —  0. 

In  Fig.  114,   inductive    external    circuit,  of   the  condition,  r/x 

=  -|-  .75,  with  a  power  factor,  .6. 

In  Fig.  115,  inductive  external  circuit,  of  the  condition,  r=  0,, 
with  a  power  factor,  0. 


§164]          ALTERNATING-CURRENT  GENERATOR. 


241 


In  Fig.  116,  external  circuit  with  leading  current,  of  the  condi- 
tion, r  /  x  =  —  .75,  with  a  power  factor,  .6. 

In  Fig.  117,  external  circuit  with  leading  current,  of  the  condi- 
tion, r  =  0,  with  a  power  factor,  0. 


20 
21 
22 
20 
18 
16 

M 

12 
10 
8 
6 

4 

J 

1 

\ 

|  FIELD  CHARA 
E72500,  Z^MOj.  i 

CTERISTIC 

\ 

\ 

7 

\ 

N 

N 

^ 

-— 

^ 

A 

\ 

X 

\ 

~o    • 

/ 

^ 

^ 

\ 

08 

y 

X 

\ 

"« 

/ 

9 

/' 

N 

\ 

/. 

> 

\> 

s 

\ 

1 

X 

"\\ 

>,  \ 

u 

/ 

\  \ 

1 

\ 

[4 

20        40        60        80        100       120       UO       160       180       200       220      240      260  Amps. 
Fig.  114.    Field  Characteristic  of  Alternator,  at  60%  Power-factor  on  Inductive  Load. 

Such  a  curve  is  called  a  field  characteristic. 

As  shown,  the  E.M.F.  curve  at  non-inductive  load  is 
nearly  horizontal  at  open  circuit,  nearly  vertical  at  short 
circuit,  and  is  similar  to  an  arc  of  an  ellipsis. 

With  reactive  load  the  curves  are  more  nearly  straight 
lines. 

The  voltage  drops  rapidly  on  inductive,  rises  on  capacity 
load. 

The  output  increases  from  zero  at  open  circuit  to  a  max- 
imum, and  then  decreases  again  to  zero  at  short  circuit. 


242 


ALTERNATING-CURRENT  PHENOMENA. 


164 


1 

\ 

FIELD  C 
£0=2500, 

JHARACTE 
Zo-1-^j,  r^ 

:RISTIC 

o,  90°  Lag 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

"o  uj 

\ 

<^ 

XS| 

s. 

11  II 

O 
00 

-x 

^ 

\ 

v^ 

\ 

V 

o  o 

^ 

N 

f   .. 

\ 

j 

/ 

\ 

\ 

\ 

/ 

\ 

\ 

\ 

y 

\ 

\ 

\ 

\ 

0 

, 

\ 

\ 

Fig.  115.    Field  Characteristic  of  Alternator,  on  Wattless  Inductive  Load. 
Volts 


1200 


400 


3000 


E0=2500 


FIELD  CHARACTERIST 


Ampe 


=  -.75  OP  60^  P. F 


200  210  £>*'    280 


>/>• 

£>*'    2 


\i 


Fig.  116.     Field  Characteristic  of  Alternator,  at  60%  Power-factor  on  Condenser  Load. 


§164] 


AL  TERN  A  TING-CURRENT  GENERA  TOR. 


243 


« 

'i 

FIE 

LD  C 

PTARACTE 

i 

RIST 

C 

I 

i 

i 

r 

E^2'pOO,|Zo=1-iOj, 
=  o,  90°  Lagging  Curre 

it 

1 

i 

i 

I*R 

=  0 

/  . 

V 

/ 

• 

/ 

/ 

/ 

i 

/ 

'    Itj 

i 

/ 

"I 

£/ 

/ 

^/ 

/ 

/ 

7 

i 
1 

/ 

\/ 

s 

***i 
O/ 
'// 

T 

>i 

J. 

r 

<?/ 

/ 

/ 

llJ 

< 

/ 

11 
st 

88 

^°l 
/ 

i 

i/ 

/ 

/ 

L 

/t 

/ 

0  0 

X    X 

yl 

j/ 

1 

/ 

/ 

3 

1 

/ 

/ 

/ 

/> 

K 

/ 

/ 

/ 

/ 

/^ 

/ 

/ 

/ 

''/. 

/ 

/ 

^ 

'/ 

^'" 

4 

'/ 

x  10 

D=Ar 

nps. 

2          ' 
Fig.  117.. 

6           8          10         12        II         10         IS         20         22        24         26 
F/'e/rf  Characteristic  of  Alternator,  on  Wattless  Condenser  Load. 

244  AL  TERN  A  TING-CURRENT  PHENOMENA.  [  §  1 65 


4oO    400    350    300 


£ 


0    100     50     0 
50 


7 


100 


100 


50     100    loO     200 


2500,  Z=1 


\ 


300    350    100   150500 


\\ 


Fig.  118.    Field  Characteristic  of  Alternator. 

165.  The  dependence  of  the  terminal  voltage,  E,  upon 
the  phase  relation  of  the  external  circuit  is  shown  in  Fig. 
119,  which  gives,  at  impressed  E.M.F., 

E0  =  2,500  volts, 
for  the  currents, 

/=  50,  100,  150,  200,  250  amperes, 

the  terminal  voltages,  E,  as  ordinates,  with  the  inductance 
factor  of  the  external  circuit, 

s 


as  abscissae. 


V; 


166.    If  the  internal  impedance  is  negligible-  compared 
with  the  external  impedance,  then,  approximately, 


E0 


+ 


§166] 


AL  TERN  A  TING-CURRENT  GENERA  TOR. 


245 


1      .9 


.7     .6      .5     .4      .3      .2     .1      0      -.1    -.2    -.3    -.4    -.5     %6     -.7    -.8 
Fig.  119.    Regulation  of  Alternator  on  Various  Loads. 


-.9    -i 


that  is,  an  alternator  with  small  internal  resistance  and  syn- 
chronous reactance  tends  to  regulate  for  constant  terminal 
voltage. 

Every  alternator  does  this  near  open  circuit,  especially 
on  non-inductive  load. 

Even  if  the  synchronous  reactance,  x0,  is  not  quite  neg- 
ligible, this  regulation  takes  place,  to  a  certain  extent,  on 
non-inductive  circuit,  since  for 

x  =  0,    E  = 


2  —  -4-    °- 
r         r* 


and  thus  the  expression  of  the  terminal  voltage,  E,  contains 
the  synchronous  reactance,  x0,  only  as  a  term  of  second 
order  in  the  denominator. 

On  inductive  circuit,  however,  x0  appears  in  the  denom- 
inator as  a  term  of  first  order,  and  therefore  constant  poten- 
tial regulation  does  not  take  place  as  well. 


246  ALTERNATING-CURRENT  PHENOMENA,        [§  167 

With  a  non-inductive  external  circuit,  if  the  synchronous 
reactance,  ;r0,  of  the  alternator  is  yery  large  compared  with 
the  external  resistance,  r, 

T      E. 1_ 

current  *  — , 

Xi 


approximately,  or  constant ;  or,  if  the  external  circuit  con 
tains  the  reactance,  x, 


approximately,  or  constant. 

The  terminal  voltage  of  a  non-inductive  circuit  is 

XQ 

approximately,  or  proportional  to  the  external  resistance. 
If  an  inductive  circuit, 

E  = —  yV2  -j-  x*  > 

dC§  ~~j~~   OC 

approximately,  or  proportional  to  the  external  impedance. 

167.  That  is,  on  a  non-inductive  external  circuit,  an 
alternator  with  very  low  synchronous  reactance  regulates 
for  constant  terminal  voltage,  as  a  constant-potential  ma- 
chine ;  an  alternator  with  a  very  high  synchronous  reac- 
tance regulates  for  a  terminal  voltage  proportional  to  the 
external  resistance,  as  a  constant-current  machine. 

Thus,  every  alternator  acts  as  a  constant-potential  ma- 
chine near  open  circuit,  and  as  a  constant-current  machine 
near  short  circuit. 

The  modern  alternators  are  generally  more  or  less  ma- 


§  167]  ALTERNATING-CURRENT  GEA?ERATOR.  247 

chines  of  the  first  class  ;  the  old  alternators,  as  built  by 
Jablockkoff,  Gramme,  etc.,  were  machines  of  the  second 
class,  used  for  arc  lighting,  where  constant-current  regula- 
tion is  an  advantage. 

Obviously,  large  external  reactances  cause  the  same  reg- 
ulation for  constant  current  independently  of  the  resistance, 
r,  as  a  large  internal  reactance,  ;r0. 

On  non-inductive  circuit,  if 

/•"O 
— _ 


the  output  is          P  =  IE  = ^- 2 

(T  +  ro)  +  XQ 

^  D  „  2  *-2_|_    *-  2 

77  :  =  {(r   '       ""   '       °~*  ^°2; 
hence,  if  XQ  = 

or  r  - 


then 

That  is,  the  power  is  a  maximum,  and 


+  r0 


and  y 


V2  s0  {%  +  /'0| 

Therefore,  with  an  external  resistance  equal  to  the  inter- 
nal impedance,  or,  r  =  ZQ  =  Vr02  -f  :r02 ,  the  output  of  an 
alternator  is  a  maximum,  and  near  this  point  it  regulates 
for  constant  output  ;  that  is,  an  increase  of  current  causes 
a  proportional  decrease  of  terminal  voltage,  and  inversely. 

The  field  characteristic  of  the  alternator  shows  this 
effect  plainly. 


248      ALTERNATING-CURRENT  PHENOMENA.     [§§168-170 


CHAPTER    XVII. 

SYNCHRONIZING    ALTERNATORS. 

168.  All  alternators,  when  brought  to  synchronism  with 
each  other,  will  operate  in  parallel  more  or  less  satisfactorily. 
This  is  due  to  the  reversibility  of  the  alternating-current 
machine  ;  that  is,  its  ability  to  operate  as  synchronous  motor. 
In  consequence  thereof,  if  the  driving  power  of  one  of  sev- 
eral parallel-operating  generators  is  withdrawn,  this  gene- 
rator will  keep  revolving  in  synchronism  as  a  synchronous 
motor ;  and  the  power  with  which  it  tends  to   remain   in 
synchronism  is  the  maximum  power  which  it  can  furnish 
as  synchronous  motor  under  the  conditions  of  running. 

169.  The  principal  and  foremost  condition  of  parallel 
operation  of   alternators  is  equality  of  frequency  ;   that  is, 
the  transmission  of  power  from  the  prime  movers  to  the 
alternators  must  be  such  as  to  allow  them  to  run  at  the 
same   frequency   without   slippage   or   excessive  strains   on 
the  belts  or  transmission  devices. 

Rigid  mechanical  connection  of  the  alternators  cannot  be 
considered  as  synchronizing ;  since  it  allows  no  flexibility  or 
phase  adjustment  between  the  alternators,  but  makes  them 
essentially  one  machine.  If  connected  in  parallel,  a  differ- 
ence in  the  field  excitation,  and  thus  the  induced  E.M.F.  of 
the  machines,  must  cause  large  cross-current ;  since  it  cannot 
be  taken  care  of  by  phase  adjustment  of  the  machines. 

Thus  rigid  mechanical  connection  is  not  desirable  for 
parallel  operation  of  alter'nators. 

170.  The  second  important  condition  of  parallel  opera- 
tion is  uniformity  of  speed ;  that  is,  constancy  of  frequency. 


§171]  SYNCHRONIZING  ALTERNATORS.  24  & 

If,  for  instance,  two  alternators  are  driven  by  independent 
single-cylinder  engines,  and  the  cranks  of  the  engines  hap- 
pen to  be  crossed,  the  one  engine  will  pull,  while  the  other 
is  near  the  dead-point,  and  conversely.  Consequently,  alter- 
nately the  one  alternator  willfctend  to  speed  up  and  the 
other  slow  down,  then  the  other  speed  up  and  the  first 
slow  down.  This  effect,  if  not  taken  care  of  by  fly-wheel 
capacity,  causes  a  " hunting"  or  pumping  action;  that  is,  a 
fluctuation  of  the  lights  with  the  period  of  the  engine  revo- 
lution, due  to  the  alternating  transfer  of  the  load  from  one 
engine  to  the  other,  which  may  even  become  so  excessive 
as  to  throw  the  machines  out  of  step:  This  difficulty  does 
not  exist  with  turbine  or  water-wheel  driving. 

171.  In  synchronizing  alternators,  we  have  to  distin- 
guish the  phenomenon  taking  place  when  throwing  the 
machines  in  parallel  or  out  of  parallel,  and  the  phenome- 
non when  running  in  synchronism. 

When  connecting  alternators  in  parallel,  they  are  first 
brought  approximately  to  the  same  frequency  and  same 
voltage  ;  and  then,  at  the  moment  of  approximate  equality 
of  phase,  as  shown  by  a  phase-lamp  or  other  device,  they 
are  thrown  in  parallel. 

Equality  of  voltage  is  much  less  important  with  modern 
alternators  than  equality  of  frequency,  and  equality  of  phase 
is  usually  of  importance  only  in  avoiding  an  instantaneous 
flickering  of  the  lights  on  the  system.  When  two  alter- 
nators are  thrown  together,  currents  pass  between  the 
machines,  which  accelerate  the  one  and  retard  the  other 
machine  until  equal  frequency  and  proper  phase  relation 
are  reached. 

With  modern  ironclad  alternators,  this  interchange  of 
mechanical  power  is  usually,  even  without  very  careful 
adjustment  before  synchronizing,  sufficiently  limited  not 
to  endanger  the  machines  mechanically  ;  since  the  cross- 
currents, and  thus  the  interchange  of  power,  are  limited 
by  self-induction  and  armature  reaction. 


250      AL  TERNA  TING-CURRENT  PHENOMENA.       [ §  172 ,  1^  3 

In  machines  of  very  low  armature  reaction,  that  is, 
machines  of  "  very  good  constant  potential  regulation," 
much  greater  care  has  to  be  exerted  in  the  adjustment 
to  equality  'of  frequency,  voltage,  and  phase,  or  the  inter- 
change of  current  may  become  so  large  as  to  destroy  the 
machine  by  the  mechanical  shock ;  and  sometimes  the 
machines  are  so  sensitive  in  this  respect  that  it  is  prefer- 
able not  to  operate  them  in  parallel.  The  same  applies 
in  getting  out  of  step. 

172.  When    running   in  synchronism,   nearly  all   types 
of  machines  will  operate  satisfactorily  ;    a  medium  amount 
of  armature  reaction  is  preferable,  however,  such  as  is  given 
by    modern     alternators  —  not     too     high     to    reduce    the 
synchronizing   power  too  much,   nor  too   low  to  make  the 
machine  unsafe  in  case  of  accident,  such  as  falling  out  of 
step,  etc. 

If  the  armature  reaction  is  very  low,  an  accident,  —  such 
as  a  short  circuit,  falling  out  of  step,  opening  of  the  field 
circuit,  etc.,  —  may  destroy  the  machine.  If  the  armature 
reaction  is  very  high,  the  driving-power  has  to  be  adjusted 
very  carefully  to  constancy  ;  since  the  synchronizing  power 
of  the  alternators  is  too  weak  to  hold  them  in  step,  and 
carry  them  over  irregularities  of  the  driving-power. 

173.  .Series  operation  of  alternators  is  possible  only  by 
rigid  mechanical   connection,   or   by  some   means  whereby 
the    machines,   with   regard   to  their  synchronizing   power, 
act  essentially  in  parallel ;  as,  for  instance,  by  the  arrange- 
ment shown  in  Fig.  120,  where  the  two  alternators,  Alf  A2, 
are  connected   in  series,   but   interlinked   by  the  two   coils 
of  a  large  transformer,    T,   of  which  the  one  is   connected 
across  the  terminals  of  one  alternator,  and  the  other  across 
the  terminals  of  the  other  alternator  in   such  a  way  that, 
when  operating  in  series,  the  coils  of  the  transformer  will 
be  without  current.      In  this  case,  by  interchange  of  power 


§174] 


SYNCHRONIZING  ALTERNATORS. 


251 


through    the    transformers,   the   series    connection   will    be 
maintained  stable. 


SULQJUUUL) 


r 


Fig.  120. 


174.    In  two  parallel  operating  alternators,  as  shown  in 
Fig.  121,  let  the  voltage  at  the  common  bus  bars  be  assumed 


Fig.   121. 


as   zero   line,   or   real   axis   of   coordinates    of   the   complex 
method  ;   and  let  — 


252  ALTERNATING-CURRENT  PHENOMENA.         [§  174r 

e    =  difference  of  potential  at  the  common  bus  bars  of 

the  two  alternators, 

Z  =  r  — jx  =  impedance  of  external  circuit, 
Y  =  g  -\-jb  =  admittance  of  external  circuit ; 

hence,  the  current  in  external  circuit  is 


r  —  jx 
Let 

El  =  e±  — /<?/  =  a2  (cos  u^  —/sin  o^)  =  induced  E.M.F.  of  first 
machine ; 

Ez  =  ^2  — /V2'  =  az  (cos  w2  — /sin  o>2)  =  induced  E.M.F.  of  sec- 
ond machine  ; 

/!    =  4  +///  =  current  of  first  machine  ; 

/2    =  /2  -{-yV/  =  current  of  second  machine  ; 

Zx  =  rt  — yX  =  internal  impedance,  and  Fx  =^t  -\-  jb^  =  inter- 
nal admittance,  of  first  machine  ;  ' 

Z2  =  r%  —  jx»  =F=  internal  impedance,  and  Yz  =  gz  -\-  jb^  =  inter- 
nal admittance,  of  second  machine. 

Then, 


=  ^  +  7^!,  or  e1  —jei  =  (e  -} 
This  gives  the  equations  — 


or  eight  equations  with  nine  variables:  elt  e^,  e2,  e2',  il, 


174]  SYNCHRONIZING  ALTERNATORS.  253 

Combining  these  equations  by  twos, 


substituted  in 

.    ,    . 

-     *1   -f*2  •=<?£•, 

we  have 

<?i£i  +  'i'£i  +  '2^2  +  e*1>*  =  c(gi  +  g*  +  g)  ; 

and  analogously, 

^1  —  ^iVl  +  ^2^2   —  ^2^2   =  *  (^1     +  ^2   +  ^)  5 

dividing, 


substituting 

g  =  v  cos  a         <?!  =  #!  cos  wj         <f2  =  a2  cos  o>2 
^  =  v  sin  a         */  =  tfj  sin  wj         ^2'  =  a2  sin  w2 
gives 

^  +  £1  +  ^2  __  a\  v\  cos  (^  —  (uQ  +  ^2  ^2  cos  (a2  —  ^2) 
/^  +  b^  -\-  b-L       a±  v^  sin  (aA  —  wj)  +  ^2  ^2  sin  (a2  —  a>2) 


as  the  equation  between  the  phase  displacement  angles 
and  o>2  in  parallel  operation. 

The  power  supplied  to  the  external  circuit  is 


of  which  that  supplied  by  the  first  machine  is, 

P\  =  «i  ; 
by  the  second  machine, 

/a  ==  «a  . 
The  total  electrical  work  done  by  both  machines  is, 

P=Pl  +  P,, 
of  which  that  done  by  the  first  machine  is, 

PI  =  *i  h  —  e{  i{  ; 
by  the  second  machine, 

P.2  =  gz  4  _  e2'  //  - 


254  ALTERNATING-CURRENT  PHENOMENA.        [§175 

The  difference  of  output  of  the  two  machines  is, 

A  p  =/i  —  /2  =  e  (h  —  4)  5 
denoting 

oij  -f-  to2  a>!  —  o>2         cj 

~^r          ~^r 

A/  /AS  may  be  called  the  synchronizing  power  of  the 
machines,  or  the  power  which  is  transferred  from  one  ma- 
chine to  the  other  by  a  change  of  the  relative  phase  angle. 

175.    SPECIAL  CASE.  —  Two  equal  alternators  of  equal 

excitation. 

al  =  a%  =  a 

z±  =  Z<L  =  z§  . 

Substituting  this  in  the  eight  initial  equations,  these 
assume  the  form,  — 


eg  =  \ 

<?/;  =  //  +  // 

tf  '+•*!•  ***  +  *{*-m-< 

Combining  these  equations  by  twos, 

el  +ez   =2e  +  e(rQg+  x^b 


e         e     =  e   x^g—  r^; 

substituting  ^  =  a  cos  WT 

^/  =  a  sin  tu! 
ez  =  a  cos  w2 
^2r  =  «  sin  o>2, 

we  have       «  (cos  w:  +  cos  oi2)  =  e  (2  +  r0<§-  +  XQ  b) 
a  (sin  A!  +  sin  o>2)  = 

expanding  and  substituting  — 

oil  -f-  o>2 


§175]  SYNCHRONIZING   ALTERNATORS.  255 


or  ,  a  cos  e  cos  8  =  e  I  1  -f-    °^  ^* — ^ 

a  sin  e  cos  8  =  e  —°— °—  ; 

»  2 

hence  tan  c  =  ->   •yo^'~*o =  constant. 


That  is  £>!  +  wa  =  constant; 


and        cos  8  -  -1  +  ^  + 

a  cos  8 


or,  ^  = 


v/(i+^^)2+(^v^) 

at  no-phase  displacement  between  the  alternators,  or, 


we  have    e  = 


| 


From  the  eight  initial  equations  we  get,  by  combina- 


subtracted  and  expanded  — 


or,  since 

el  —  <?2  =  <?  (cos  oj!  —  cos  w2)  =  —  2  ^2  sin  e  sin  8 
e{  —  el  =  a  (sin  w1  —  sin  w2)  =       2  #  cos  e  sin  8  ; 

we  have 

2  ^  sin  8  (  .      -, 

*i  —  z2  = 2 —  fe«  cos  c  ~  r°  sm  €} 

=  2  ^JFO  sin  8  cos  (c  +  a), 
where 

tan  a  =  — -  . 

fo 


256  ALTEltNATING-CURRENT  PHENOMENA.         [§175 

The  difference  of  output  of  the  two  alternators  is 


hence,  substituting, 
substituting, 


a  cos  8 


1    i 


sin  e  = 


COS  e  = 


1       I     '  0  X    1~  ^0  {/   \      i     /   ^06    —  '0"  \ 

2         J       \         2         j 
i  -j — ^ — - 


i  +  /o*  7"ot/  + 
T  \ 

we  have, 


.  /^     ,    ?o,f  + VA        ;   ATO^  -  r0/A  ) 
o(  1  +  -— 2— -     ~^o(  --17--)      ; 

A/= 


+  -L-2*-)  +  (*0*  2   ^ 
expanding, 


or 


A  p  = 


Hence,  the  transfer  of  power  between  the  alternators, 
A/,  is  a  maximum,  if  8  =  45°  ;  or  wt  —  oi2  =  90°  ;  that  is, 
when  the  alternators  are  in  quadrature. 


§176]  SYNCHRONIZING   ALTERNATORS.  257 

The   synchronizing   power,    A  p  /  A  8,  is  a  maximum  if 
8=0;  that  is,  the  alternators  are  in  phase  with  each  other. 

176.    As  an  instance,  curyes  may  be  plotted 
for, 

a    =  2500, 


with  the  angle  8  =  a>1       M2  as  abscissae,  giving  . 

the  value  of  terminal  voltage,  e ; 

the  value  of  current  in  the  external  circuit,  /  =  ey ; 

the  value  of   interchange  of   current  between  the  alternators, 

*  i  —  h  5 
the  value  of  interchange  of  power  between  the  alternators,  A  p 

=A-A; 

the  value  of  synchronizing  power,  — ^  ,  in  dash-dot  line,  Curve  V. 

A  o 

For  the  condition  of  external  circuit, 

g  =  0,  b  =  0,  y  =  0, 

.05,  0,  .05, 

.08,  0,  .08, 

.03,  +  .04,  .05, 

.03,  -  .04,  .05. 


258  ALTERNATING-CURRENT  PHENOMENA.       [§177 


CHAPTER    XVIiL 

SYNCHRONOUS    MOTOR. 

177.  In  the  chapter  on  synchronizing  alternators  we 
have  seen  that  when  an  alternator  running  in  synchronism 
is  connected  with  a  system  of  given  E.M.F.,  the  work  done 
by  the  alternator  can  be  either  positive  or  negative.  In 
the  latter  case  the  alternator  consumes  electrical,  and 
consequently  produces  mechanical,  power ;  that  is,  runs 
as  a  synchronous  motor,  so  that  the  investigation  of  the 
synchronous  motor  is  already  contained  essentially  in  the 
equations  of  parallel-running  alternators. 

Since  in  the  foregoing  we  have  made  use  mostly  of 
the  symbolic  method,  we  may  in  the  following,  as  an 
instance  of  the  graphical  method,  treat  the  action  of  the 
synchronous  motor  diagrammatically. 

Let  an  alternator  of  the  E.M.F.,  Elt  be  connected  as 
synchronous  motor  with  a  supply  circuit  of  E.M.F.,  EQ, 
by  a  circuit  of  the  impedance  Z. 

If  EQ  is  the  E.M.F.  impressed  upon  the  motor  termi- 
nals, Z  is  the  impedance  of  the  motor  of  induced  E.M.F., 
El.  If  EQ  is  the  E.M.F.  at  the  generator  terminals,  Z  is 
the  impedance  of  motor  and  line,  including  transformers 
and  other  intermediate  apparatus.  If  EQ  is  the  induced 
E.M.F.  of  the  generator,  Z  is  the  sum  of  the  impedances 
of  motor,  line,  and  generator,  and  thus  we  have  the  prob- 
lem, generator  of  induced  E.M.F.  EQ,  and  motor  of  induced 
E.M.F.  El ;  or,  more  general,  two  alternators  of  induced 
E.M.Fs.,  EQ,  Elt  connected  together  into  a  circuit  of  total 
impedance,  Z. 

Since  in  this  case  several  E.M.Fs.  are  acting  in  circuit 


§177]  SYNCHRONOUS  MOTOR.  259 

with  the  same  current,  it  is  convenient  to  use  the  current, 
/,  as  zero  line  OI  of  the  polar  diagram. 

If  /  =  i  =  current,  and  Z  =  impedance,  r  =  effective 
resistance,  x  =  effective  reaqtance,  and  z  =  Vr2  +  x*  = 
absolute  value  of  impedance,  then  the  E.M.F.  consumed 
by  the  resistance  is  El  =  ri,  and  in  phase  with  the  cur- 
rent, hence  represented  by  vector  OEl;  and  the  E.M.F. 
consumed  by  the  reactance  is  £2  =  xi,  and  90°  ahead  of 
the  current,  hence  the  E.M.F.  consumed  by  the  impedance 
is  E  =  ^(Evf  +  (£j2,  or  =  i  Vr2  -f-  x^  =  iz,  and  ahead  of 
the  current  by  the  angle  8,  where  tan  5  =  x  /  r. 

We  have  now  acting  in  circuit  the  E.M.Fs.,  E,  El,  EQ't 
or  El  and  E  are  components  of  EQ  ;  that  is,  EQ  is  the 
diagonal  of  a  parallelogram,  with  El  and  E  as  sides. 

Since  the  E.M.Fs.   Elt  E2,  E,  are  represented  in   the 


diagram,  Fig.  122,  by  the  vectors  OElt  OE2,  OE,  to  get 
the  parallelogram  of  EQ,  El,  E,  we  draw  arc  of  circle 
around  0  with  EQt  and  around  E  with  E^.  Their  point 
of  intersection  gives  the  impressed  E.M.F.,  OEQ  =  EQ, 
and  completing  the  parallelogram  OE,  EQ,  Elt  we  get 
OEl  =  Elt  the  induced  E.M.F.  of  the  motor. 


IOEQ  is  the  difference  of  phase  between  current  and  im- 
pressed E.M.F.,  or  induced  E.M.F.  of  the  generator. 

IOE^  is  the  difference  of  phase  between  current  and  in- 

duced E.M.F.  of  the  motor. 

And  the  power  is  the  current  /times  the  projection  of  the  E.M.F. 
upon  the  current,  or  the  zero  line  OL 


Hence,  dropping  perpendiculars,  EQE^  and  E1E1\  from 
Q  and  El  upon  Of,  it  is  — 


PQ  =  i  X  OE^  =  power  supplied  by  induced  E.M.F.  of  gen- 
erator. 

PI  =  i  X  O£il  =  electric  power  transformed  in  mechanical 
power  by  the  motor. 

P  =  i  x  OEl  =  power  consumed  in  the  circuit  by  effective 
resistance. 


260 


ALTERNATING-CURRENT  PHENOMENA.        [§178 


Obviously  P0  =  Pl  +  P. 

Since  the  circles  drawn  with  EQ  and  E1  around  O  and  E 
respectively  intersect  twice,  two  diagrams  exist.  In  gen- 
eral, in  one  of  these  diagrams  shown  in  Fig.  122  in  drawn 


Fig.   122. 

lines,  current  and  E.M.F.  are  in  the  same  direction,  repre- 
senting mechanical  work  done  by  the  machine  as  motor. 
In  the  other,  shown  in  dotted  lines,  current  and  E.M.F.  are 
in  opposite  direction,  representing  mechanical  work  con- 
sumed by  the  machine  as  generator. 

Under  certain  conditions,  however,  EQ  is  in  the  same,  E± 
in  opposite  direction,  with  the  current ;  that  is,  both  ma- 
chines are  generators. 

178.  It  is  seen  that  in  these  diagrams  the  E.M.Fs.  are 
considered  from  the  point  of  view  of  the  motor ;  that  is, 


§178] 


SYNCHRONOUS  MOTOR. 


261 


work  done  as  synchronous  motor  is  considered  as  positive, 
work  done  as  generator  is  negative.  In  the  chapter  on  syn- 
chronizing generators  we  took  the  opposite  view,  from  the 
generator  side.  ,  ^ 

In  a  single  unit-power  transmission,  that  is,  one  generator 
supplying  one  synchronous  motor  over  a  line,  the  E.M.F. 
consumed  by  the  impedance,  E  —  OE,  Figs.  123  to  125,  con- 
sists of  three  components ;  the  E.M.F.  OE%  —  Ez,  consumed 


Fig.   123. 


by  the  impedance  of  the  motor,  the  E.M.F.  E^  E%  =  Ez 
consumed  by  the  impedance  of  the  line,  and  the  E.M.F. 
E%  E  =  E±  consumed  by  the  impedance  of  the  generator. 
Hence,  dividing  the  opposite  side  of  the  parallelogram  £1EQy 
in  the  same  way,  we  have  :  OEl  =  El  =  induced  E.M.F.  of 
the  motor,  OEZ  =  E^  =  E.M.F.  at  motor  terminals  or  at 
end  of  line,  OE%  =  _E3  =  E.M.F.  at  generator  terminals, 
or  at  beginning  of  line.  OEQ  =  E^  =  induced  E.M.F.  of 
generator. 


262 


ALTERNA  TING-CURRENT  PHENOMENA. 


179 


The  phase  relation  of  the  current  with  the  E.M.Fs.  EI} 
)  depends  upon  the  current  strength  and  the  E.M.Fs.  E1 


and  EQ. 


179.  Figs.  123  to  125  show  several  such  diagrams  for 
different  values  of  El9  but  the  same  value  of  /  and  EQ. 
The  motor  diagram  being  given  in  drawn  line,  the  genera- 
tor diagram  in  dotted  line. 


Fig.  124. 

As  seen,  for  small  values  of  El  the  potential  drops  in 
the  alternator  and  in  the  line.  For  the  value  of  E1  =  EQ 
the  potential  rises  in  the  generator,  drops  in  the  line,  and 
rises  again  in  the  motor.  For  larger  values  of  Elt  the 
potential  rises  in  the  alternator  as  well  as  in  the  line,  so 
that  the  highest  potential  is  the  induced  E.M.F.  of  the 
motor,  the  lowest  potential  the  induced  E.M.F.  of  the  gen- 
erator. 


§  ISO] 


SYNCHRONOUS  MOTOR. 


It  is  of  interest  now  to  investigate  how  the  values  of 
these  quantities  change  with  a  change  of  the  constants. 


Fig.   125. 


180.    A.  —  Constant  impressed  E.M.F.  EQ,  constant  current 
strength  I  =  z,  variable  motor  excitation  E±.     (Fig.  126.) 


If  the  current  is  constant,  =  z;  OE,  the  E.M.F.  con- 
sumed by  the  .impedance,  and  therefore  point  E,  are  con- 
stant. Since  the  intensity,  but  not  the  phase  of  EQ  is 
constant,  EQ  lies  on  a  circle  <?0  with  EQ  as  radius.  From 
the  parallelogram,  OE,  EQ  El  follows,  since  E1EQ  parallel 
and  =  OE,  that  E1  lies  on  a  circle  e1  congruent  to  the  circle 
eQ,  but  with  E{,  the  image  of  E,  as  center  :  OE{  =  OE. 

We  can  construct  now  the  variation  of  the  diagram  with 
the  variation  of  E^ ;  in  the  parallelogram  OE  EQ  E-±,  O  and 
E  are  fixed,  and  EQ  and  El  move  on  the  circles  eQ  e1  so  that 
EQ  El  is  parallel  to  OE. 


264 


A  L  TERN  A  TING-CURRENT  PHENOMENA.        [  §  1 8O 


The  smallest  value  of  El  consistent  with  current  strength 


7  is 


=  E     01 


In  this  case  the  power  of    the 


motor  is  01  j1  x  /,  hence  already  considerable.  Increasing 
El  to  Q2lt  03-p  etc.,  the  impressed  E.M.Fs.  move  to  02,  03, 
€tc.,  the  power  is  /  x  02^,  /  x  03^,  etc.,  increases  first, 


Fig.   126. 


reaches  the  maximum  at  the  point  3U  3,  the  most  extreme 
point  at  the  right,  with  the  impressed  E.M.F.  in  phase  with 
the  current,  and  then  decreases  again,  while  the  induced 
E.M.F.  of  the  motor  El  increases  and  becomes  =  EQ  at 
4j,  4.  At  51?  5,  the  power  becomes  zero,  and  further  on 
negative  ;  that  is,  the  motor  has  changed  to  a  dynamo,  and 


§  ISO]  SYNCHRONOUS  MOTOR.  265 

produces  electrical  energy,  while  the  impressed  E.M.F.  E^ 
still  furnishes  electrical  energy,  that  is,  both  machines  as 
generators  feed  into  the  line,  until  at  61?  6,  the  power  of  the 
impressed  E.M.F.  EQ  becomes  zero,  and  further  on  power 
begins  to  flow  back ;  that  is,  tfce  motor  is  changed  to  a  gen- 
erator and  the  generator  to  a  motor,  and  we  are  on  the 
generator  side  of  the  diagram.  At  7L,  7,  the  maximum  value 
of  Elt  consistent  with  the  current  C,  has  been  reached,  and 
passing  still  further  the  E.M.F.  El  decreases  again,  while 
the  power  still  increases  up  to  the  maximum  at  8n  8,  and 
then  decreases  again,  but  still  El  remaining  generator,  E§ 
motor,  until  at  11^  11,  the  power  of  EQ  becomes  zero ;  that 
is,  EQ  changes  again  to  a  generator,  and  both  machines  are 
generators,  until  at  12t,  12,  where  the  power  of  El  is  zero, 
El  changes  from  generator  to  motor,  and  we  come  again  to 
the  motor  side  of  the  diagram,  and  while  E±  still  decreases, 
the  power  of  the  motor  increases  until  lu  1,  is  reached. 

Hence,  there  are  two  regions,  for  very  large  El  from 
6  to  7,  and  for  very  small  E^  from  11  to  12,  where  both 
machines  are  generators  ;  otherwise  the  one  is  generator, 
the  other  motor. 

For  small  values  of  El  the  current  is  lagging,  begins, 
however,  at  2  to  lead  the  induced  E.M.F.  of  the  motor  E19 
at  3  the  induced  E.M.F.  of  the  generator  EQ. 

It  is  of  interest  to  note  that  at  the  smallest  possible 
value  of  Ev  lp  the  power  is  already  considerable.  Hence, 
the  motor  can  run  under  these  conditions  only  at  a  certain 
load.  If  this  load  is  thrown  off,  the  motor  cannot  run  with 
the  same  current,  but  the  current  must  increase.  We  have 
here  the  curious  condition  that  loading  the  motor  reduces, 
unloading  increases,  the  current  within  the  range  between 
1  and  12. 

The  condition  of  maximum  output  is  3,  current  in  phase 
writh  impressed  E.M.F.  Since  at  constant  current  the  loss 
is  constant,  this  is  at  the  same  time  the  condition  of  max- 
imum efficiency :  no  displacement  of  phase  of  the  impressed 


266 


ALTERNATING-CURRENT  PHENOMENA.        [§181 


E.M.F.,  or  self-induction  of  the  circuit  compensated  by  the 
effect  of  the  lead  of  the  motor  current.  This  condition  of 
maximum  efficiency  of  a  circuit  we  have  found  already  in 
Chapter  VIII.  on  Inductance  and  Capacity. 


181.        B. 


Q  and  E1  constant,  I  variable. 


Obviously  £Q  lies  again  on  the  circle  e0  with  EQ  as  radius 
and  O  as  center. 


Fig.  127. 


E  lies  on  a  straight  line  e,  passing  through  the  origin. 

Since  in  the  parallelogram  OE  EQ  Ev  EEQ  =  Elt  we 
derive  EQ  by  laying  a  line  EE$  =  E^  from  any  point  E 
in  the  circle  eQ,  and  complete  the  parallelogram. 

All  these  lines  EEQ  envelop'  a  certain  curve  elt  which 


§181] 


SYNCHRONOUS  MOTOR. 


267 


can  be  considered  as  the  characteristic  curve  of  this  prob- 
lem, just  as  circle  el  in  the  former  problem. 

These  curves  are  drawn  in  Figs.  127,  128,  129,  for  the 
three  cases  :  1st,  El  =  EQ  ;,2d,  E1  <  EQ ;  3d,  El  >EQ. 

In  the  first   case,   El  =  Jj^  (Fig.  127),  we  see  that  at 


Fig.  128. 

very  small  curren.,  that  is  very  small  OE,  the  current  / 
leads  the  impressed  E.M.F.  EQ  by  an  angle  E0OC  =  o»0. 
This  lead  decreases  with  increasing  current,  becomes  zero, 
and  afterwards  for  larger  current,  the  current  lags.  Taking 
now  any  pair  of  corresponding  points  E,  EQ,  and  producing 

until  it  intersects  eif  in  E{  we  have  ^^  Ei  OE  =  90°, 

r/f0,  thus:  OE1  =  EEQ=OEQ  =  . 


L  ;  that  is,  EEi 


268 


ALTERNA  TING-CURRENT  PHENOMENA. 


181 


That  means  the  characteristic  curve  ^  is  the  enve- 
lope of  lines  EEi ,  of  constant  lengths  2  E^ ,  sliding  between 
the  legs  of  the  right  angle  E{  OE\  hence,  it  is  the  sextic 
hypocyloid  osculating  circle  eQ,  which  has  the  general  equa- 
tion, with  e,  e{  as  axes  of  coordinates  : 


V>2  =  V  4  £Q2 

In  the  next  case,  E^  <  E^  (Fig.  128)  we  see  first,  that 
the  current  can   never  become  zero  like  in  the  first  case,. 


Fig.  129. 

El  =  EQ,  but  has  a  minimum  value  corresponding  to  the 

E    —  E 
minimum  value  of   OE^ :   //=  -    — ,  and  a   maximum 

value  :  //  =  • —         — — .    Furthermore,  the  current  can  never 

Jo 

lead  the  impressed  E.M.F.  E^9  but  always  lags.      The  mini- 


§  182]  SYNCHRONOUS  MOTOR.  269 

mum  lag  is  at  the  point  H.  The  locus  ev  as  envelope  of  the 
lines  EE§,  is  a  finite  sextic  curve,  shown  in  Fig.  128. 

In  the  case  El  >  EQ  (Fig.  129)  the  current  cannot  equal 
zero  either,  but  begins  at  a?  finite  value  C^,  corresponding 

E   —  E 
to  the  minimum  value  of  OE^  :   //  =  —          — .     At  this 

value  however,  the  alternator  El  is  still  generator  and 
changes  to  a  motor,  its  power  passing  through  zero,  at  the 
point  corresponding  to  the  vertical  tangent,  onto  ev  with 
a  very  large  lead  of  the  impressed  E.M.F.  against  the  cur- 
rent. At  H  the  lead  changes  to  lag. 

The  minimum  and   maximum   value   of   current   in   the 
three  conditions  are  given  by  : 

1st.    7=0,  7=^. 

z 

*>H      /  —  -^o  —  Ev  T  _  EQ  +  E\ 

-U.      2    -  —  2    -  


OJ          r  «J   —  -^0  r          -L^Q  "f  f*I 

OU.      Y    —  —  5  Y    —   -     , 

Z  Z 

Since  the  current  passing  over  the  line  at  El  =  O,  that 
is,  when  the  motor  stands  still,  is  70  =  E^j  zt  we  see  that 
in  such  a  synchronous  motor-plant,  when  running  at  syn- 
chronism, the  current  can  rise  far  beyond  the  value  it  has 
at  standstill  of  the  motor,  to  twice  this  value  at  1,  some- 
what less  at  2,  but  more  at  3. 

Hence  in  such  a  case,  if  the  synchronous  motor  drops 
out  of  step,  the  current  passing  over  the  line  goes  down  to 
one-half  or  less  ;  or,  in  other  words,  in  such  a  system  the 
motor,  under  certain  conditions  of  running,  is  more  liable 
to  burn  up  than  when  getting  out  of  step. 

182.    C.    EQ  =  constant,  Ev  varied  so  that  the  efficiency  is  a 
maximum  for  all  currents.     (Fig.  130.) 

Since  we  have  seen  that  the  output  at  a  given  current 
strength,  that  is,  a  given  loss,  is  a  maximum,  and  therefore 


270 


AL  TERN  A  TING-CURRENT  PHENOMENA. 


182 


the  efficiency  a  maximum,  when  the  current  is  in  phase 
with  the  induced  E.M.F.  EQ  of  the  generator,  we  have  as 
the  locus  of  E0  the  point  E0  (Fig.  130),  and  when  E  with 
increasing  current  varies  on  *?,  El  must  vary  on  the  straight 
line  el  parallel  to  e. 

Hence,  at  no-load  or  zero  current,  El  =  EQ,  decreases 
with  increasing  load,  reaches  a  minimum  at  OEf  perpen- 
dicular to  elt  and  then  increases  again,  reaches  once  more 


Fig.   130. 

El  =  EQ  at  Ef,  and  then  increases  beyond  EQ.  The  cur- 
rent is  always  ahead  of  the  induced  E.M.F.  El  of  the  motor, 
and  by  its  lead  compensates  for  the  self-induction  of  the 
system,  making  the  total  circuit  non-inductive. 

The  power  is  a  maximum  at  Ef,  where  OEf  =  EfEQ  = 
1/2  x  02f0,  and  is  then  =  /  x  EO/%.  Hence,  since  OE*  = 
Ir  =  EQ/2,  1=  EQ/2randP  =  E02/4:r,  hence  =  the  maxi- 
mum power  which,  over  a  non-inductive  line  of  resistance  r 
can  be  transmitted,  at  50  per  cent,  efficiency,  into  a  non- 
inductive  circuit. 


183]  SYNCHRONOUS  MOTOR. 

In  this  case, 


271 


In  general,  it  is,  taken  fgpm  the  diagram,  at  the  condi- 
tion of  maximum  efficiency : 


Comparing  these  results  with  those  in  Chapter  IX.  on 
Self-induction  and  Capacity,  we  see  that  the  condition  of 
maximum  efficiency  of  the  synchronous  motor  system  is 
the  same  as  in  a  system  containing  only  inductance  and 
capacity,  the  lead  of  the  current  against  the  induced  E.M.F. 
EI  here  acting  in  the  same  way  as  the  condenser  capacity 
in  Chapter  IX. 


Fig.   131. 


183.          D.    EQ  =  constant;  P  =  constant. 

If  the  power  of  a  synchronous  motor  remains  constant, 
we  have   (Fig.  131)  /  x  OEJ-  =  constant,  or,  since  OE1  = 


272 


A  L  TERN  A  TING-CURRENT  PHENOMENA . 


183 


Ir,    I  =  OEl/r,    and:     OE1  x  OE?  =  OE1  x  ElEQl  = 
constant. 

Hence  we  get  the  diagram  for  any  value  of  the  current 
/,  at  constant  power  P^ ,  by  making  OE1  =  Ir,  E1EJ*  =  P1/  C 
erecting  in  EQl  a  perpendicular,  which  gives  two  points  of, 
intersection  with  circle  eQt  EQ,  one  leading,  the  other  lagging. 
Hence,  at  a  given  impressed  E.M.F.  EQ,  the  same  power  Pl 


E,  | 

1250  7 

1100/1580  3116.7 

1480  32 

1050/1840  2/25 


2120 
2170 


37.5 

40 

45.5 


Ef  1000 

P=IOOO 

46  <  E^2200 


2/25 

22 

3/16.7 


Fig.   132. 


can  be  transmitted  by  the  same  current  I  with  two  different 


induced  E.M.Fs.  E}  of  the  motor;  one, 


=  EEQ  small, 


corresponding  to  a  lagging  current  ;  and  the  other,  OEl  = 
EEQ  large,  corresponding  to  a  leading  current.  The  former 
is  shown  in  dotted  lines,  the  latter  in  drawn  lines,  in  the 
diagram,  Fig.  131. 

Hence  a  synchronous  motor  can  work  with  a  given  out- 
put, at  the  same  current  with  two  different  counter  E.M.Fs. 


§  183] 


SYNCHRONOUS  MOTOR. 


273 


£19  and  at  the  same  counter  E.M.F.  Elt  at  two  different 
currents  /.  In  one  of  the  cases  the  current  is  leading,  in 
the  other  lagging.  ^, 

In  Figs.  132  to  135  are  shown  diagrams,  giving  the  points 

E0  =  impressed  E.M.F.,  assumed  as  constant  =  1000  volts, 
E  —  E.M.F.  consumed  by  impedance, 
E'  =  E.M.F.  consumed  by  resistance. 


1450  17.3 

1170/1910     10/30 
1040/1930     8/37.5 


E=1000 

P=6000 

340 <  E,<1920 

7<  I  <  43 


Fig,  133. 


of  the  motor,  £lf  is  O£l}  equal  and 
shown    in   the   diagrams,    to   avoid 


The  counter  E.M.F. 
parallel  EEQ,  but  not 
complication. 

The  four  diagrams  correspond  to  the  values  of  power, 
or  motor  output, 
P  =  1,000,       6,000, 


9,000, 


P  =    1.000  46  <  E  <  2,200, 

P  =    6.000  340  <  El  <  1,920, 

P  =    9,000  540  <El<  1,750, 

P  =  12,000  920  <  El  <  1,320,     20     <  I  <  30 


12,000  watts,  and  give  : 

1  <  7  <  49          Fig.  132. 
7  <  /  <  43          Fig.  133. 
Fig.  134. 


11.8  <  /  <  38.2 


Fig.  153. 


274 


AL  TERNA  TING-CURRENT  PHENOMENA. 


183 


E,      I 

1440  21.2 

1200/1660  15/30 

1080/1750  13/34.7 


1          900/1590      11.8/38. 


ft'    720/1100   13/34.7 
6//     620/820   15/30 
540      21.2 


E0=1000 

P=9000 

540 <  E,<1750 

11. 8<  l<38.2 


Fig.  134. 


Ei  I 

1280  24.5 

1120/1320  21/28.6 

---/1260  20/30 


920/1100         21/28.6 
24.5 


E^IOOO 

P  =  I2000 

920<  Ef  1320 

20<l<30 


Fig.  135. 


As  seen,  the  permissible  value  of  counter  E.M.F.  El  and 
of  current  /,  becomes  narrower  with  increasing  output. 


§  184]  SYNCHRONOUS  MOTOR.  275 

In  the  diagrams,  different  points  of  E^  are  marked  with 
1,  2,  3  .  .  .,  when  corresponding  to  leading  current,  with 
21,  31,  .  .  . ,  when  corresponding  to  lagging  current. 

The  values  of  counter  £.M.F.  E±  and  of  current  I  are 
noted  on  the  diagraras,  opposite  to  the  corresponding  points 

In  this  condition  it  is  interesting  to  plot  the  current  as 
function  of  the  induced  E.M.F.  El  of  the  motor,  for  con- 
stant power  Pl.  Such  curves  are  given  in  Fig.  139  and 
explained  in  the  following  on  page  278. 

184.  While  the  graphic  method  is  very  convenient  to 
get  a  clear  insight  into  the  interdependence  of  the  different 
quantities,  for  numerical  calculation  it  is  preferable  to  ex- 
press the  diagrams  analytically. 

For  this  purpose, 

Let  z  =  V/'2  +  x'2  =  impedance  of  the  circuit  of  (equivalent) 
resistance  r  and  (equivalent)  reactance  x  =  2  TT  NL,  containing 
the  impressed  E.M.F.  <?0*  and  the  counter  E.M.F.  ^  of  the  syn- 
chronous motor;  that  is,  the  E.M.F.  induced  in  the  motor  arma- 
ture by  its  rotation  through  the  (resultant)  magnetic  field. 

Let  i  =  current  in  the  circuit  (effective  values). 

The  mechanical  power  delivered  by  the  synchronous 
motor  (including  friction  and  core  loss)  is  the  electric 
power  consumed  by  the  C. E.M.F.  el ;  hence  — 

thus,  — 

cos 

*<?, 

(2) 


sn 


,0  =  t/1  _ 


*  If  e^  =  E.M.F.  at  motor  terminals,  z  =  internal  impedances  of  the 
motor;  if  e$  =  terminal  voltage  of  the  generator,  z  =  total  impedance  of  line 
and  motor;  if  eQ  =  E.M.F.  of  generator,  that  is,  E.M.F.  induced  in  generator 
armature  by  its  rotation  through  the  magnetic  field,  z  includes  the  generator 
impedance  also. 


276  ALTERNATING-CURRENT  PHENOMENA.        [§184 

The  displacement  of  phase  between  current  i  and  E.M.F. 
e  =  z  i  consumed  by  the  impedance  z  is  : 

cos  (*>)  =  - 

.     ... 
siri  (t  e) 


Since  the  three  E.M.Fs.  acting  in  the  closed  circuit : 

eQ  =  E.M.F.  of  generator, 

el  =  C.E.M.F.  of  synchronous  motor, 

e  =  zi=  E.M.F.  consumed  by  impedance, 

form  a  triangle,  that  is,  e1  and  e  are  components  of  <?0,  it  is 
(Fig.  136) : 


^2   e1  ^2  ^  2  _  ^,  2  _      2  /2 

hence,      cos  (elfe)  =  -* — — -1-      —  =  Ji — —*—     - ,  (5) 

Zi  6 1  C  £  Z  2  €^ 

since,  however,  by  diagram  : 

cos  (el ,  e)  =  cos  (/,  e  —  i,  e{) 

=  cos  (t,  e)  cos  (/,  ^)  +  sin  (/,  e)  sin  (i,  e^)        (6) 

substitution  of  (2),  (3)  and  (5)  in  (6)  gives,  after  some  trans- 
position : 


the  Fundamental  Equation  of  the  Synchronous  Motor,  relat- 
ing impressed  E.M.F.,  e0 ;  C. E.M.F.,  ^  ;  current  z;  power, 
/,  and  resistance,  r ;  reactance,  x ;  impedance  #. 

This  equation  shows  that,  at  given  impressed  E.M.F.  eQ, 
and  given  impedance  s  =  Vr2  -j-  «^2>  three  variables  are  left, 
^j,  /,  /,  of  which  two  are  independent.  Hence,  at  given  e^ 
and  z,  the  current  i  is  not  determined  by  the  load  /  only, 
but  also  by  the  excitation,  and  thus  the  same  current  i  can 
represent  widely  different  loads  /,  according  to  the  excita- 
tion ;  and  with  the  same  load,  the  current  i  can  be  varied 
in  a  wide  range,  by  varying  the  field  excitation  el. 

The  meaning  of  equation  (7)  is  made  more  perspicuous 


§184] 


SYNCHRONOUS  MOTOR. 


277 


by  some  transformations,  which  separate  el  and  z,  as  .func- 
tion of/  and  of  an  angular  parameter  <£. 
Substituting  in  (7)  the  new  coordinates : 


(8) 


we  get 


substituting  again, 


F/g.   73(5. 


.   737. 


we  get 


(9) 


-„  V2  -  2r    =  2       e 


a  -  a  V2  -  e  b  =  V(l  -  e2)  (2  a2  -  2  j32  -  P),          (11) 

and,  squared, 

2  2 

substituting 

ca 


(a  -  e  /;)  V2  _ 


/?  Vl  -  e2  =  a/, 
gives,  after  some  transposition, 


(13) 


(14) 


278  ALTERNATING-CURRENT  PHENOMENA.        [§184 

hence,  if  r- 

R  =     /(I  -<*)(<*-  2*6)  a  .       (15) 

V'  2c2 


the  equation  of  a  circle  with  radius  R. 

Substituting  now  backwards,  we  get,  with  some  trans- 
positions : 

{r*  (e*  +  *2/2)  -  z*  (e*  -  2  r/)}2  +  {r  x  (e*  -  *2/2)}2  = 

^V(^02-4r/)  (17) 

the  Fundamental  Eqtiation  of  the  SyncJironous  Motor  in  a 
modified  form. 

The  separation  of  el  and  i  can  be  effected  by  the  intro- 
duction of  a  parameter  <j>  by  the  equations: 


r x  (e±   —  z2 i 2)  =  x z e^  V^02  —  krp  sin  9 
These  equations  (18),  transposed,  give 

— °( -cos4>  +  sinc^  W— ^rP 


(        / 


The  parameter  9  has  no  direct  physical  meaning,  appar- 
ently. 

These  equations  (19)  and  (20),  by  giving  the  values  of 
^  and  i  as  functions  of  /  and  the  parameter  9  enable  us 
to  construct  the  Power  Characteristics  of  the  Synchronous 
Motor,  as  the  curves  relating  ^  and  z,  for  a  given  power  /> 
by  attributing  to  9  all  different  values. 


§  185]  SYNCHRONOUS  MOTOR.  279 

Since  the  variables.?'  and  iv  in  the  equation  of  the  circle 
(16)  are  quadratic  functions  of  e1  and  i,  the  Power  Charac- 
teristics of  tJie  SyncJironous  Motor  are  Quartic  Curves. 

They  represent  the  action,  of  the  synchronous  motor 
under  all  conditions  of  load  and  excitation,  as  an  element 
of  power  transmission  even  including  the  line,  etc. 

Before  discussing  further  these  Power  Characteristics,, 
some  special  conditions  may  be  considered. 

185.  A.     Maximum   Output. 

Since  the  expression  of  ^  and  i  [equations  (19)  and 
(20)]  contain  the  square  root,  V^02  —  4r/,  it  is  obvious 
that  the  maximum  value  of  /  corresponds  to  the 9  moment 
where  this  square  root  disappears  by  passing  from  real  to- 
imaginary  ;  that  is, 

e*  _  4  rp  =  0, 


This  is  the  same  value  which  represents  the  maximum 
power  transmissible  by  E.M.F.,  eQ,  over  a  non-inductive  line 
of  resistance,  r\  or,  more  generally,  the  maximum  power 
which  can  be  transmitted  over  a  line  of  impedance, 

z  =  -Vr2  +  x2, 

into  any  circuit,  shunted  by  a  condenser  of  suitable  capacity.. 
Substituting  (21)  in  (19)  and  (20),  we  get, 


(22) 

and  the  displacement  of  phase  in  the  synchronous  motor. 

/        -\         p         r 
cos  (e1 ,  / )  =  ^_  =  -  ; 

te\      z 

hence, 

/  -\  X  /OQY 

tan  (*!,  i)  — ,  (Z6), 


280  ALTERNATIATG-CURRENT  PHENOMENA.        [§186 

that  is,  the  angle  of  internal  displacement  in  the  synchron- 
ous motor  is  equal,  but  opposite  to,  the  angle  of  displace- 
ment of  line  impedance, 

fa,  0  =  -  (',  0, 

=  -  (*,  r\  (24) 

and  consequently, 

(25) 


that   is,   the    current,   it   is    in   phase  with    the    impressed 
E.M.F.,  *0. 

If  5  <  2  r,  ^i  <  ^0;  that  is,  motor  E.M.F.  <  generator  E.M.F. 
If  z  =  2  r,  el  =  e0  ;  that  is,  motor  E.M.F.  =  generator  E.M.F. 
If  z  >  2  r,  el  >  <?0;  that  is,  motor  E.M.F.  >  generator  E.M.F. 

In  either  case,  the  current   in  the   synchronous  motor   is 
leading. 

186.  B.     Running  Light,  p  =  0. 

When  running  light,  or  for  p  =  0,  we  get,  by  substitut- 
ing in  (19)  and  (20), 


+  - 

2  (          z 


(26) 


Obviously  this  condition  can  never  be  fulfilled  absolutely, 
since/  must  at  least  equal  the  power  consumed  by  friction, 
etc. ;  and  thus  the  true  no-load  curve  merely  approaches  the 
curve/  =  0,  being,  however,  rounded  off,  where  curve  (26) 
gives  sharp  corners. 

Substituting  /  =  0  into  equation  (7)  gives,  after  squar- 
ing and  transposing, 

f*  +  e(?  +  W4  -  2  eftf  -  2  z*i2e<?  +  2  ra/V  -  2  *2/V  =  0.  (27) 

This  quartic  equation  can  be  resolved  into  the  product 
of  two  quadratic  equations, 

e\   +  •s2/2  —  e^  -j-  2  xie\  =  0  generator. 
e*  -_  02^-2  _     2  _  2  .         =  0  motor. 


§  186]  SYNCHRONOUS  MOTOR.  281 

which  are  the  equations  of  two  ellipses,  the  one  the  image 
of  the  other,  both  inclined  with  their  axes. 

The  minimum  value  of  C.E.M.F.,  elt  is  el  =  0  at  /  =  ^.   (IP) 

»    "  z 

The  minimum  value  of  currenj,  z,  is  /  =  0  at  eL  =  eQ  .         (30) 

The  maximum  value  of  E.M.F.,  ely  is  given  by  Equation  (28),. 
/=  e*  _|_  z*i*  _  ^  _t  2  */>,  =  0  ; 

by  the  condition, 

*/^i  d  f  I  dl  f\  0  •     i  rv 

"7^  =  ~~  ~77TT  =  °»  as  z  l  ±  xe^  =  °' 
dt  df  I  dei 

hence, 

i  =  e»—,     ^i  =  =F^o-  (31) 

rsr  r 

The  maximum  value  of  current,  i,  is  given  by  equation 
(28)  by 


at        A 

-—  =  0,  as 


(32> 


If,  as  abscissae,  elf  and  as  ordinates,  zi,  are  chosen,  the 
axis  of  these  ellipses  pass  through  the  points  of  maximum 
power  given  by  equation  (22). 

It  is  obvious  thus,  that  in  the  curves  of  synchronous 
motors  running  light,  published  by  Mordey  and  others,  the 
two  sides  of  the  V-shaped  curves  are  not  straight  lines,  as 
usually  assumed,  but  arcs  of  ellipses,  the  one  of  concave,  the 
other  of  convex,  curvature. 

These  two  ellipses  are  shown  in  Fig.  138,  and  divide  the 
whole  space  into  six  parts  —  the  two  parts  A  and  A  '  ,  whose 
areas  contain  the  quartic  curves  (19)  (20)  of  synchronous 
motor,  the  two  parts  B  and  B1,  whose  areas  contain  the 
quartic  curves  of  generator,  and  the  interior  space  C  and 
exterior  space  D,  whose  points  do  not  represent  any  actual 
condition  of  the  alternator  circuit,  but  make  elf  i  imaginary. 

A  and  A'  and  the  same  B  and  !>'  ,  are  identical  condi- 
tions of  the  alternator  circuit,  differing  merely  by  a  simul- 


282 


ALTERNA TING-CURRENT  PHENOMENA. 


187 


200- 


160- 


\ 


r.000       4000       3000X^2000       loop 


Volt3  1000          2000\/ 3000          4000  5000 


\ 


/A' 


\ 


B' 


F/g.  738. 

taneous  reversal  of  current  and  E.M.F.  ;  that  is,  differing 
by  the  time  of  a  half  period. 

Each  of  the  spaces  A  and  B  contains  one  point  of  equa- 
tion (22),  representing  the  condition  of  maximum  output 
of  generator,  viz.,  synchronous  motor. 

187.    C.    Minimum   Current  at  Given  Power. 

The  condition  of  minimum  current,  i,  at  given  power,  /, 
is  determined  by  the  absence  of  a  phase  displacement  at  the 
impressed  E.M.F.  eQ, 


-:§  187]  SYNCHRONOUS  MOTOR.  283 

This  gives  from  diagram  Fig.  132, 

^  =  ^  +  /V>_2*>0^  (33) 

>    '  z 

or,  transposed, 

e,  *=  V(*o-'>)a  +  'a*2.  (34) 

This  quadratic  curve  passes  through  the  point  of  zero 
current  and  zero  power, 

/  =  0,  e\  =  e*, 

through  the  point  of  maximum  power  (22), 

/_   ^  _e*z 

1  —  7T~  >  *i  —  77~  > 

2  r  2  r 

and  through  the  point  of  maximum  current  and  zero  power, 

,,     _  eQX  /QPIN 

<?i  =  -  ,  (OO) 


and  divides  each  of  the  quartic  curves  or  power  character- 
istics into  two  sections,  one  with  leading,  the  other  with 
lagging,  current,  which  sections  are  separated  by  the  two 
points  of  equation  34,  the  one  corresponding  to  minimum, 
the  other  to  maximum,  current. 

It  is  interesting  to  note  that  at  the  latter  point  the 
current  can  be  many  times  larger  than  the  current  which 
would  pass  through  the  motor  while  at  rest,  which  latter 
current  is, 

/-"*.;  (36) 

2 

while  at  no-load,  the  current  can  reach  the  maximum  value, 

i  -a,  '  :.;.;"      (35) 

the  same  value  as  would  exist  in  a  non-inductive  circuit  of 
the  same  resistance. 

The  minimum  value  at  C.E.M.  F.  e^,  at  which  coincidence 


284  ALTERNATING-CURRENT  PHENOMENA.       [§  188 

of  phase  (^0,  i)  =  0,  can  still  be  reached,  is  determined  from 
equation  (34)  by, 

de\  =  Q . 

di  ' 
as 

f=**2     *i  =  'oj.  (37) 

The  curve  of  no-displacement,  or  of  minimum  current,  is 
shown  in  Figs.  138  and  139  in  dotted  lines.* 


188.      D.    Maximum  Displacement  of  Phase. 

(eOJ  i)  =  maximum. 
At  a  given  power/  the  input  is, 

A  =/  +  i*r  =  <?„/  cos  (*0,  /)  ;  (38) 

hence, 

cos(,?0,  i)  =  £±_i!r.  (39) 

<V 

At  a  given  power  /,  this  value,  as  function  of  the  current 
i,  is  a  maximum  when 


//     .<?<,* 

this  gives, 

/  =  /V;  (40) 

or, 


That  is,  the  displacement  of  phase,  lead  or  lag,  is  a 
maximum,  when  the  power  of  the  motor  equals  the  power 

*  It  is  interesting  to  note  that  the  equation  (34)  is  similar  to  the  value, 
^  =  V(/0  —  z>)2  —  z'2.r2,  which  represents  the  output  transmitted  over  an 
inductive  line  of  impedance,  z  =  vV'2  -f  jr2  into  a  non-inductive  circuit. 

Equation  (34)  is  identical  with  the  equation  giving  the  maximum  voltage, 
^1  ,  at  current,  z,  which  can  be  produced  by  shunting  the  receiving  circuit  with  a 
condenser;  that  is,  the  condition  of  "  complete  resonance  "  of  the  line,  z  — 

VV2  +  jr2,  with  current,  /.  Hence,  referring  to  equation  (35),  e^  =  <?0  ~~  is 
the  maximum  resonance  voltage  of  the  line,  reached  when  closed  by  a  con- 
denser of  reactance,  —  jc. 


188] 


SYNCHRONOUS  MOTOR. 


285 


consumed  by  the  resistance ;  that  is,  at  the  electrical  effi- 
ciency of  50  per  cent. 

Substituting  (40)  in  equation  (7)  gives,  after  squaring 


600          1000 


8000.         '2000        5000         ooOO        4000         4500         6000         5500 
Fig.  139. 


and  transposing,  the  Quartic  Equation  of    Maximum  Dis- 
placement, 

(e2  -  e2)2  +  /V  (z2  +  8  r2)  +  2  /  V  (5  r2  -  z2)  -  2  i2e2 

<V  +  3  r2)  =  0.  (42) 

The  curve  of  maximum  displacement  is  shown  in  dash- 
dotted  lines  in  Figs.  138  and  189.      It  passes  through  the 


286     ALTERNATING-CURRENT  PHENOMENA.     [§§189,190 

point  of  zero  current  —  as  singular  or  nodal  point  —  and 
through  the  point  of  maximum  power,  where  the  maximum 
displacement  is  zero,  and  it  intersects  the  curve  of  zero 
displacement. 

189.  E.    Constant  Counter  E.M.R 

At  constant  C.E.M.F.,  ^  =  constant, 

If 


the  current  at  no-load  is  not  a  minimum,  and  is  lagging. 
With  increasing  load,  the  lag  decreases,  reaches  a  mini- 
mum, and  then  increases  again,  until  the  motor  falls  out  of 
step,  without  ever  coming  into  coincidence  of  phase. 


If 


the  current  is  lagging  at  no  load ;  with  increasing  load  the 
lag  decreases,  the  current  comes  into  coincidence  of  phase 
with  eQ ,  then  becomes  leading,  reaches  a  maximum  lead  ; 
then  the  lead  decreases  again,  the  current  comes  again  into 
coincidence  of  phase,  and  becomes  lagging,  until  the  motor 
falls  out  of  step. 

If  <?0  <  el ,  the  current  is  leading  at  no  load,  and  the 
lead  first  increases,  reaches  a  maximum,  then  decreases ; 
and  whether  the  current  ever  comes  into  coincidence  of 
phase,  and  then  becomes  lagging,  or  whether  the  motor 
falls  out  of  step  while  the  current  is  still  leading,  depends, 
whether  the  C.E.M.F.  at  the  point  of  maximum  output  is 
>  ^Oor  <  *0. 

190.  F.    Numerical  Instance. 

Figs.  138  and  139  show  the  characteristics  of  a  100- 
kilowatt  motor,  supplied  from  a  2500-volt  generator  over  a 
distance  of  5  miles,  the  line  consisting  of  two  wires,  No. 
2  B.  &  S.G.,  18  inches  apart. 


§  19O]  SYNCHRONOUS  MOTOR.  287 

In  this  case  we  have, 

<?0  =  2500  volts  constant  at  generator  terminals ; 
r  =      10  ohms,  including  line  and  motor; 
x  =      20  ohms,  including  line  and  motor ; 
hence  z  =     22.36  ohms. 

Substituting  these  values,  we  get, 


25002  -  ^  -  500  i  2  -  20/  =  40  V*V  -/2  (7) 

2  +  500  *a  -  31.25  X  106  +  100  /}2  +  {2  e*  -  1000  *  2}2  = 

7.8125  x  1015  -  5  +  109/.  (17) 

=  5590  (19) 


3.2  X  10-6/)  +  (.894  cos  <£+  .447sin  <£)  Vl-6.4xlO~6/}. 
/  =  559  (20) 


Maximum  output, 

/  =  156.25  kilowatts  (21) 

at  <?!  =  2,795  volts 

Running  light, 


(22} 
i  =  125  amperes  ' 


e*  +  500  /a  -  6.25  X  104  =f  40  i  e,  =  0  )  ,28 

*!  =  20  /  i  V6.25  X  104  -  100  /a      | 

At  the  minimum  value  of  C.E.M.F.  el  =  0  is  /  =  112  (29) 
At  the  minimum  value  of  current,  i  =  0  is  el  =  2500  (30) 
At  the  maximum  value  of  C.E.M.F.  el  =  5590  is  /  =  223.5  (31) 
At  the  maximum  value  of  current  /  =  250  is  el  =  5000  (32) 

Curve  of  zero  displacement  of  phase, 


=  10  V(250  -  O2  +  4  /a  (34) 


=  10  V6.25  x  104  -  500  /  +  5  / a 
Minimum  C.E.M.F.  point  of  this  curve, 

i  =  50         e1  =  2240  (35) 

Curve  of  maximum  displacement  of  phase, 

/  =  10  i2  (40) 

(6.25  X  106-^2)2  +  .65  x  106 1*  -  1010/2  =  0.        (42) 


288  ALTERNATING-CURRENT  PHENOMENA.       [§  191 

Fig.  138  gives  the  two  ellipses  of  zero  power,  in  drawn 
lines,  with  the  curves  of  zero  displacement  in  dotted,  the 
curves  of  maximum  displacement  in  dash-dotted  lines,  and 
the  points  of  maximum  power  as  crosses. 

Fig.  139  gives  the  motor-power  characteristics,  for, 

/  =  10  kilowatts. 
p  =  50  kilowatts. 
/  ==  100  kilowatts. 
/  =  150  kilowatts. 
/  =  156.25  kilowatts. 

together  with  the  curves  of  zero  displacement,  and  of  maxi- 
mum displacement. 


191.  G.    Discussion  of  Results. 

The  characteristic  curves  of  the  synchronous  motor,  as 
shown  in  Fig.  139,  have  been  observed  frequently,  with 
their  essential  features,  the  V-shaped  curve  of  no  load,  with 
the  point  rounded  off  and  the  two  legs  slightly  curved,  the 
one  concave,  the  other  convex ;  the  increased  rounding  off 
and  contraction  of  the  curves  with  increasing  load  ;  and 
the  gradual  shifting  of  the  point  of  minimum  current  with 
increasing  load,  first  towards  lower,  then  towards  higher, 
values  of  C.E.M.F.  el. 

The  upper  parts  of  the  curves,  however,  I  have  never 
been  able  to  observe  experimentally,  and  consider  it  as 
probable  that  they  correspond  to  a  condition  of  synchro- 
nous motor-running,  which  is  unstable.  The  experimental 
observations  usually  extend  about  over  that  part  of  the 
curves  of  Fig.  139  which  is  reproduced  in  Fig.  140,  and  in 
trying  to  extend  the  curves  further  to  either  side,  the  motor 
is  thrown  out  of  synchronism. 

It  must  be  understood,  however,  that  these  power  char- 
acteristics of  the  synchronous  motor  in  Fig.  1 39  can  be  con- 
sidered as  approximations  only,  since  a  number  of  assump- 


191] 


SYNCHRONOUS  MOTOR. 


289 


tions  are  made  which  are   not,  or  only  partly,  fulfilled  in 
practice.     The  foremost  of  these  are : 

1.  It  is  assumed  that  e±  can  be  varied  unrestrictedly, 
while  in  reality  the  possible  increase  of  el  is  limited  by 
magnetic  saturation-.  Thus*  in  Fig.  139,  at  an  impressed 
E.M.F.,  <?0  =  2,500  volts,  el  rises  up  to  5,590  volts,  which 
may  or  may  not  be  beyond  that  which  can  be  produced 
by  the  motor,  but  certainly  is  beyond  that  which  can  be 
constantly  given  by  the  motor. 


HO 


J 


TBO 


\ 


Volts 


500 


1600' 


.2000     3500     300.0 
Fig.  140. 


3300 


5000 


2.  The  reactance,  x,  is  assumed  as  constant.  While 
the  reactance  of  the  line  is  practically  constant,  that  of  the 
motor  is  not,  but  varies  more  or  less  with  the"  saturation, 
decreasing  for  higher  values.  This  decrease  of  x  increases 
the  current  z,  corresponding  to  higher  values  of  elt  and 
thereby  bends  the  curves  upwards  at  a  lower  value  of  el 
than  represented  in  Fig.  139. 

It  must  be  understood  that  the  motor  reactance  is  not 
.a  simple  quantity,  but  represents  the  combined  effect  of 


290  ALTERNATING-CURRENT  PHENOMENA.        [§191 

self-induction,  that  is,  the  E.M.F.  induced  in  the  armature 
conductor  by  the  current  flowing  therein  and  armature 
reaction,  or  the  variation  of  the  C. E.M.F.  of  the  motor 
by  the  change  of  the  resultant  field,  due  to  the  superposi- 
tion of  the  M.M.F.  of  the  armature  current  upon  the  field 
excitation  ;  that  is,  it  is  the  "  synchronous  reactance." 

3.  These  curves  in  Fig.  139  represent  the  conditions 
of  constant  electric  power  of  the  motor,  thus  including  the 
mechanical  and  the  magnetic  friction  (core  loss).  While 
the  mechanical  friction  can  be  considered  as  approximately 
constant,  the  magnetic  friction  is  not,  but  increases  with 
the  magnetic  induction  ;  that  is,  with  ^ ,  and  the  same  holds 
for  the  power  consumed  for  field  excitation. 

Hence  the  useful  mechanical  output  of  the  motor  will 
on  the  same  curve,  p  =  const.,  be  larger  at  points  of  lower 
C. E.M.F.,  elf  than  at  points  of  higher  el;  and  if  the  curves 
are  plotted  for  constant  useful  mechanical  output,  the  whole 
system  of  curves  will  be  shifted  somewhat  towards  lower 
values  of  e1 ;  hence  the  points  of  maximum  output  of  the 
motor  correspond  to  a  lower  E.M.F.  also. 

It  is  obvious  that  the  true  mechanical  power-character- 
istics of  the  synchronous  motor  can  be  determined  only 
in  the  case  of  the  particular  conditions  of  the  installation 
under  consideration. 


192,193]  COMMUTATOR  MOTORS.  291 


CHAPTER    XIX. 


COMMUTATOR   MOTORS. 

192.  Commutator   motors  —  that   is,   motors   in  which 
the    current   enters   or   leaves    the   armature   over   brushes 
through    a    segmental    commutator  —  have    been    built    of 
various   types,   but    have    not   found    any   extensive    appli- 
cation, in  consequence  of  the  superiority  of  the  induction 
and    synchronous   motors,   due  to   the  absence  of   cbmmu- 
tators. 

The  main  subdivisions  of  commutator  motcrs  are  the 
repulsion  motor,  the  series  motor,  and  the  shunt  motor. 

REPULSION    MOTOR. 

193.  The    repulsion    motor   is   an   induction   motor   or 
transformer  motor ;   that   is,   a   motor   in   which   the   main 
current   enters  the  primary   member   or   field   only,   while 
in   the   secondary  member,  or   armature,   a   current   is   in- 
duced, and  thus  the  action  is  due  to  the  repulsive  thrust 
between  induced  current  and  inducing  magnetism. 

As  stated  under  the  heading  of  induction  motors,  a 
multiple  circuit  armature  is  required  for  the  purpose  of 
having  always  secondary  circuits  in  inductive  relation  to 
the  primary  circuit  during  the  rotation.  If  with  a  single- 
coil  field,  these  secondary  circuits  are  constantly  closed 
upon  themselves  as  in  the  induction  motor,  the  primary 
circuit  will  not  exert  a  rotary  effect  upon  the  armature 
while  at  rest,  since  in  half  of  the  armature  coils  the  cur- 
rent is  induced  so  as  to  give  a  rotary  effort  in  the  one 
direction,  and  in  the  other  half  the  current  is  induced  to 


292 


ALTERNATING-CURRENT  PHENOMENA.        [§193 


give  a  rotary  effort   in  the   opposite  direction,   as   shown 
by  the  arrows  in  Fig.  141. 

In  the  induction  motor  a  second  magnetic  field  is  used 
to  act  upon  the  currents  induced  by  the  first,  or  inducing 
magnetic  field,  and  thereby  cause  a  rotation.  That  means 
the  motor  consists  of  a  primary  electric  circuit,  inducing 


F— - 


Fig.  141. 

in  the  armature  the  secondary  currents,  and  a  primary 
magnetizing  circuit  producing  the  magnetism  to  act  upon 
the  secondary  currents. 

In  the  polyphase  induction  motor  both  functions  of  the 
primary  circuit  are  usually  combined  in  the  same  coils  ;  that 
is,  each  primary  coil  induces  secondary  currents,  and  pro- 
duces magnetic  flux  acting  upon  secondary  currents  induced 
by  another  primary  coil. 


§194] 


COMMUTATOR   MOTORS. 


293 


194.  In  the  repulsion  motor  the  difficulty  due  to  the 
equal  and  opposite  rotary  efforts,  caused  by  the  induced 
armature  currents  when  acted  upon  by  the  inducing  mag- 
netic field,  is  overcome  by 'having  the  armature  coils  closed 
upon  themselves,  either  on  short  circuit  or  through  resist- 
ance, only  in  that  position  where  the  induced  currents  give 


Fig.   142. 

a  rotary  effort  in  the  desired  direction,  while  the  armature 
coils  are  open-circuited  in  the  position  where  the  rotary 
effort  of  the  induced  currents  would  be  in  opposition  to 
the  desired  rotation.  This  requires  means  to  open  or  close 
the  circuit  of  the  armature  coils  and  thereby  introduces  the 
commutator. 

Thus  the  general  construction  of  a  repulsion  motor  is 
as  shown  in  Figs.  142  and  143  diagrammatically  as  bipolar 


294 


AL  TERN  A  TING-CURRENT  PHENOMENA.        [  §  1 94r 


motor.  The  field  is  a  single-phase  alternating  field  F,  the 
armature  shown  diagrammatically  as  ring  wound  A  consists 
of  a  number  of  coils  connected  to  a  segmental  commutator 
C,  in  general  in  the  same  way  as  in  continuous-current  ma- 
chines. Brushes  standing  under  an  angle  of  about  45°  with 
the  direction  of  the  magnetic  field,  short-circuit  either  a 


\ 


Fig.  143. 

part  of  the  armature  coils  as  shown  in  Fig.  142,  or  the 
whole  armature  by  a  connection  from  brush  to  brush  as 
shown  in  Fig.  143. 

The  former  arrangement  has  the  disadvantage  of  using  a 
part  of  the  armature  coils  only.  The  second  arrangement 
has  the  disadvantage  that,  in  the  passage  of  the  brush  from 
segment  to  segment,  individual  armature  coils  are  short- 


§  195] 


COMMUTATOR   MOTORS. 


295- 


circuited,  and  thereby  give  a  torque  in  opposite  direction  to 
the  torque  developed  by  the  main  induced  current  flowing 

through  the  whole  armature  from  brush  to  brush. 

> 

195.  Thus  the  repulsion^motor  consists  of  a  primary 
electric  circuit,  a  magnetic  circuit  interlinked  therewith,, 
and  a  secondary  circuit  closed  upon  itself  and  displaced  in 


Fig.  144. 

space  by  45°  —  in  a  bipolar  motor  —  from  the  direction  of 
the  magnetic  flux,  as  shown  diagrammatically  in  Fig.  144. 

This  secondary  circuit,  while  set  in  motion,  still  remains 
in  the  same  position  of  45°  displacement,  with  the  magnetic 
flux,  or  rather,  what  is  theoretically  the  same,  when  moving 
out  of  this  position,  is  replaced  by  other  secondary  circuits 
entering  this  position  of  45°  displacement. 

For  simplicity,  in  the  following  all  the  secondary  quan- 


296  ALTERNATING-CURRENT  PHENOMENA.         [§196 

titles,  as  E.M.F.,  current,  resistance,  reactance,  etc.,  are 
assumed  as  reduced  to  the  primary  circuit  by  the  ratio  of 
turns,  in  the  same  way  as  done  in  the  chapter  on  Induction 
Motors. 

196.   Let 

4>  =•  maximum  magnetic  flux  per  field  pole  ; 

e  =   effective  E.M.F.  induced  thereby  in  the  field  turns;  thus  : 


e  n   =  number  of  turns, 

JV=  frequency. 
thus: 


-^-xnN 

The  instantaneous  value  of  magnetism  is 
<f)  =  <|>  sin  /?  ; 

and  the  flux  interlinked  with  the  armature  circuit 
</>!  =  3>  sin  (3,  sin  A  ; 

when  A  is  the  angle  between  the  plane  of  the  armature  coil 
and  the  direction  of  the  magnetic  flux. 

The  E.M.F.  induced  in  the  armature  circuit,  of  n  turns, 
as  reduced  to  primary  circuit,  is  thus  : 


=  —  n  3>   )  sin  A  cos  (3  —i—  +  sin  /?  cos  A  —  ( 
(  dt  dt  ) 


10 


-8 


If  N  =  frequency  in  cycles  per  second,  N^  —  speed  in 
cycles  per  second  (equal  revolutions  per  second  times  num- 
ber of  pairs  of  poles),  it  is  : 

Illf    \    f  ->•"•  .-U ••..• .": ..  i; 


§197]  COMMUTATOR  MOTORS.  297 

and   since   X  =  45°,   or   sin  X  =  cos  X  =  1/V2,   it    is,    sub- 
stituted : 


or,  since 


el=  —  e  {  cos  (3  -{-  k  sin  /?  }  ; 

where  ^  =  A  =  ratio     sPeed     ; 

N  frequency 

or  the  effective  value  of  secondary  induced  E.M.F., 


197.  Introducing  now  complex  quantities,  and  counting 
the  time  from  the  zero  value  of  rising  magnetism,  the  mag- 
netism is  represented  by 

/*; 
the  primary  induced  E.M.F., 

E  =-e; 
the  secondary  induced  E.M.F.  : 


V2 
hence,  if 

Zx  =  /i  —j'xi  =  secondary  impedance  reduced  to  primary 

circuit, 

Z   =  r  —  j  x   =  primary  impedance, 
Y  =  g  +  j  b    =  primary  admittance, 

it  is, 

secondary  current, 


^i  V2    n- 

primary  exciting  current, 


298  AL  TERN  A  TING-CURRENT  PHENOMENA.        [  §  198 

hence,  total  primary  current, 

or 

(*  V2     > 

Primary  impressed  E.M.F., 

or 


Neglecting  in  £Q  the  last  term,  as  of  higher  order, 


(  V2        ^i-7^i> 

or,  eliminating  imaginary  quantities, 


V2 


198.    The    power   consumed    by    the    primary   counter 
E.M.F.,  e,  that  is,  transferred  into  the  secondary  circuit,  is 


1  +  j 


V2  n  —  y^i 

or,  eliminating  the  imaginary  quantities, 


J— 

V2 


The  power  consumed  by  the  secondary  resistance  is 


2        r?  +  oc*   ' 

Hence,  the  difference,  or  the  mechanical  power  at  the 
motor  shaft  — 


2  (r?  -f  Xf) 


§198] 


COMMUTATOR  MOTORS. 


299 


and,  substituting  for  e, 


p  = 


{r,  (V2  -  1)  -f  kXl  V2  - 


(r  +  r:  V2 


(x 


V2  - 


If  r  and  rx  are  "small  compared  with  x  and  ^  ,  this  is 
approximately, 


Thus  the  power  is  a  maximum  for  dP  jdk  =  0,  that  is, 
_*  +  *,V2 


1000 

^,  — 

_. 

900 

/ 

^^ 

"""""""'"'' 

300 

^ 

£ 

/ 

700 

I 

/ 

/ 

R 

IPL 

LS 

ON 

M 

3TC 

)R 

COO 
500 

o 

en 

/ 

r 

e0= 

OC 

nJ 

/ 

/ 

r  = 

.1 

r,= 

05 

400 

^ 

/ 

Xs 

2. 

xi= 

300 

/ 

P  = 

1UU 
(1  7 

E? 

02  ^ 
K) 

1.4)1  K 
+  (3.14 

—  .0 
-1 

'K^ 

200 

/ 

ieo/ 

K- 

Spe 

-"rp 

ed 

3UfiJ 

— 

/ 

.2 

.4 

.G 

.8 

1. 

0 

1. 

2 

!. 

1 

1. 

6 

!. 

8 

2. 

0 

r.   745.     Repulsion  Motor. 


As  an  instance  is  shown,  in  Fig.  145,  the  power  output 
as  ordinates,  with  the  speed  k  =  A\  /  N  as  abscissae,  of  a 
repulsion  motor  of  the  constants, 

en  =  100. 


r  =-    .1          r,  =    .05 
x  ==  2.0         Xl  =  1.0 


giving  the  power, 


==  10,000  {.02  +  1.411  -  .05  I2} 

(.171  +  2£)2+  (3.14-  ,\Kf 


300 


AL  TERN  A  TING-CURRENT  PHENOMENA.          [§199 


SERIES    MOTOR.       SHUNT    MOTOR. 

199.  If,  in  a  continuous-current  motor,  series  motor  as 
well  as  shunt  motor,  the  current  is  reversed,  the  direction 
of  rotation  remains  the  same,  since  field  magnetism  and 
armature  current  have  reversed  their  sign,  and  their  prod- 


Fig.  146.     Series  Motor. 

uct,  the  torque,  thus  maintained  the  same  sign.  There- 
fore such  a  motor,  when  supplied  by  an  alternating  current, 
will  operate  also,  provided  that  the  reversals  in  field  and 
in  armature  take  place  simultaneously.  In  the  series  motor 
this  is  necessarily  the  case,  the  same  current  passing  through 
field  and  through  armature. 

With  an  alternating  current  in  the  field,  obviously  the 


§199]  COMMUTATOR  MOTORS.  301 

magnetic  circuit  has  to  be  laminated  to  exclude  eddy  cur- 
rents. 

Let,  in  a  series  motor,  Fig.  146, 
i 

3>    =  effective  magnetism  per  £ole, 

n     =  number  of  field  turns  per  pole  in  series, 

n±    =  number  of  armature  turns  in  series  between  brushes, 

/    =  number  of  poles, 

(R    =  magnetic  reluctance  of  field  circuit, 

(Rj  =  magnetic  reluctance  of  armature  circuit, 

<&!  =  effective    magnetic    flux    produced    by   armature    current 

(cross  magnetization), 
r     —  resistance    of    field    (effective    resistance,    including    hys- 

teresis), 
t\    =  resistance  of  armature  (effective  resistance,  including  hys- 

teresis), 

N  =  frequency  of  alternations, 
JV-L  =  speed  in  cycles  per  second. 

It  is  then, 

E.M.F.   induced  in    armature    conductors   by  their  rotation 
through  the  magnetic  field  (counter  E.M.F.  of  motor). 

£    =  4;z17Vi<l>10-8 
E.M.F.  of  self-induction  of  field, 
Er  =  2 


E.M.F.  of  self-induction  of  armature, 

E{  =  27r/;1^<l>110-8, 
E.M.F.  consumed  by  resistance, 

Er  =  (r  +  n)  I, 
where 

/  =  current  passing  through  motor,  in  amperes  effective. 

Further,  it  is  : 
Field  magnetism  : 

«  7  108 


302  ALTERNATING-CURRENT  PHENOMENA.         [§199 

Armature  magnetism  : 


Substituting  these  values, 
,-,         4  n 


Thus  the  impressed  E.M.F., 


rf  +  (Er 


or,  since 

*2 

=  reactance  of  field  ; 
=  2  TT  N —  =  reactance  of  armature  ; 


and 

° 


§  20O]  COMMUTATOR  MOTORS.  303 

200.    The  power  output  at  armature  shaft  is, 

P=  El 

knn^N,  F2 
—^^ 


5— , 


2   n, 


pn  N 


-  ^  §^  +  ^+ 

TT  pn  N 

The  displacement  of  phase  between  current  and  E.M.F. 
is 


2  «i_  N^x       r 
te  pn  N  * 

Neglecting,  as  approximation,  the  resistances  r  +  rlt  it  is, 


tan  o>  = 


2    «_t 


2  «i.  ^   , 

TT pn  N         2    n± 
TT  pn 


304  ALTERNATING-CURRENT  PHENOMENA.        [§201 

hence  a  maximum  for, 


2    n^  N±  =  \_ 
ic  pn  N  ~''    2 


pn  N 


or,  „         2    n     ' 

~TT     P^l 

substituting  this  in  tan  £,  it  is  : 

tan  o>  =  1,         or,         w  =  45°. 

201.    Instance  of  such  an  alternating-current  motor, 

£9  =  100         AT  =  60         /  =  2. 

r  -  .03  T!  =  .12 

x  =  .9  jq  =  .5 

«  =  10  %  =  48 

Special  provisions  were  made  to  keep  the  armature  re- 
actance a  minimum,  and  overcome  the  distortion  of  the 
field  by  the  armature  M.M.F.,  by  means  of  a  coil  closely 
surrounding  the  armature  and  excited  by  a  current  of  equal 
phase  but  opposite  direction  with  the  armature  current 
(Eickemeyer).  Thereby  it  was  possible  to  operate  a  two- 
circuit,  96-turn  armature  in  a  bipolar  field  of  20  turns,  at 
a  ratio  of 

armature  ampere-turns  r>  A 

field  ampere-turns 

It  is  in  this  case, 

7  =  100 

~  V(.023  NI  +  .15)2  +  1.96 

P=  2307V1 

(.023  JVi  -f  .15)2  +  1.96 

1.4  „  .023  TV,  -f  .15 

tan"=.023^  +  .15'0r'C°S"- 


§202] 


COMMUTATOR  MOTORS. 


305 


In  Fig.  147  are  given,  with  the  speed  TVj  as  abscissae, 
the  values  of  current  /,  power  P,  and  power  factor  cos  w 
of  this  motor. 


Amp. 


Watts 


SER 


Cos  03  = 


30  40  50  60  |70 


ES 


60 


MOTOR 


00 


A/C023 


A/IP23 


A/CQ23 


'3D 


15)' 


1.96 


1.9 


15)2+1.9 


Fig.  147.    Series  Motor. 

202.  The  shunt  motor  with  laminated  field  will  not 
operate  satisfactorily  in  an  alternating-current  circuit.  It 
will  start  with  good  torque,  since  in  starting  the  current  in 
armature,  as  well  as  in  field,  are  greatly  lagging,  and  thus 
approximately  in  phase  with  each  other.  With  increasing 
speed,  however,  the  counter  E.M.F.  of  the  armature  should 
be  in  phase  with  the  impressed  E.M.F.,  and  thereby  the 
armature  current  lag  less,  to  represent  power.  Since  how- 
ever, the  field  current,  and  thus  the  field  magnetism,  lag 
nearly  90°,  the  induced  E.M.F.  of  the  armature  will  lag 
nearly  90°,  and  thus  not  represent  power. 


306         ALTERNATING-CURRENT  PHENOMENA.  [§202 

Hence,  to  make  a  shunt  motor  work  on  alternating-cur- 
rent circuits,  the  magnetism  of  the  field  should  be  approxi- 
mately in  phase  with  the  impressed  E.M.F.,  that  is,  the  field 
reactance  negligible.  Since  the  self-induction  of  the  field  is 
far  in  excess  to  its  resistance,  this  requires  the  insertion  of 
negative  reactance,  or  capacity,  in  the  field. 

If  the  self-induction  of  the  field  circuit  is  balanced  by 
capacity,  the  motor  will  operate,  provided  that  the  armature 
reactance  is  low,  and  that  in  starting  sufficient  resistance 
is  inserted  in  the  armature  circuit  to  keep  the  armature 
current  approximately  in  phase  with  the  E.M.F.  Under 
these  conditions  the  equations  of  the  motor  will  be  similar 
to  those  of  the  series  motor. 

However,  such  motors  have  not  been  introduced,  due  to 
the  difficulty  of  maintaining  the  balance  between  capacity 
and  self-induction  in  the  field  circuit,  which  depends  upon 
the  square  of  the  frequency,  and  thus  is  disturbed  by  the 
least  change  of  frequency. 

The  main  objection  to  both  series  and  shunt  motors  is 
the  destructive  sparking  at  the  commutator  due  to  the  in- 
duction of  secondary  currents  in  those  armature  coils  which 
pass  under  the  brushes.  As  seen  in  Fig.  146,  with  the 
normal  position  of  brushes  midway  between  the  field  poles, 
the  armature  coil  which  passes  under  the  brush  incloses  the 
total  magnetic  flux.  Thus,  in  this  moment  no  E.M.F.  is 
induced  in  the  armature  coil  due  to  its  rotation,  but  the 
E.M.F.  induced  by  the  alternation  of  the  magnetic  flux 
has  a  maximum  at  this  moment,  and  the  coil,  when  short- 
circuited  by  the  brush,  acts  as  a  short-circuited  secondary 
to  the  field  coils  as  primary  ;  that  is,  an  excessive  current 
flows  through  this  armature  coil,  which  either  destroys  it, 
or  at  least  causes  vicious  sparking  when  interrupted  by  the 
motion  of  the  armature. 

To  overcome  this  difficulty  various  arrangements /have 
been  proposed,  but  have  not  found  an  application. 


§203]  COMMUTATOR   MOTORS.  307 

203.  Compared  with  the  synchronous  motor  which  has 
practically  no  lagging  currents,  and  the  induction  motor 
which  reaches  very  high  power  factors,  the  power  factor  of 
the  series  motor  is  low,  as  s.een  from  Fig.  147,  which  repre- 
sents about  the  best  possible  design  of  such  motors. 

In  the  alternating-series  motor,  as  well  as  in  the  shunt 
motor,  no  position  of  an  armature  coil  exists  wherein  the 
coil  is  dead;  but  in  every  position  E.M.F.  is  induced  in  the 
armature  coil :  in  the  position  parallel  with  the  field  flux  an 
E.M.F.  in  phase  with  the  current,  in  the  position  at  right 
angles  with  the  field  flux  an  E.M.F.  in  quadrature  with  the 
current,  intermediate  E.M.Fs.  in  intermediate  positions. 
At  the  speed  TT  N /  2  the  two  induced  E.M.Fs.  in  phase  and 
in  quadrature  with  the  current  are  equal,  and  the  armature 
coils  are  the  seat  of  a  complete  system  of  symmetrical  and 
balanced  polyphase  E.M.Fs.  Thus,  by  means  of  stationary 
brushes,  from  such  a  commutator  polyphase  currents  could 
be  derived. 


308  ALTERNATING-CURRENT  PHENOMENA.         [§204 


CHAPTER    XX. 

REACTION  MACHINES. 

204.  In  the  chapters  on  Alternating-Current  Genera- 
tors and  on  Induction  Motors,  the  assumption  has  been 
made  that  the  reactance  x  of  the  machine  is  a  constant. 
While  this  is  more  or  less  approximately  the  case  in  many 
alternators,  in  others,  especially  in  machines  of  large  arma- 
ture reaction,  the  reactance  x  is  variable,  and  is  different  in 
the  different  positions  of  the  armature  coils  in  the  magnetic 
circuit.  This  variation  of  the  reactance  causes  phenomena 
which  do  not  find  their  explanation  by  the  theoretical  cal- 
culations made  under  the  assumption  of  constant  reactance. 

It  is  known  that  synchronous  motors  of  large  and 
variable  reactance  keep  in  synchronism,  and  are  able  to 
do  a  considerable  amount  of  work,  and  even  carry  under 
circumstances  full  load,  if  the  field-exciting  circuit  is 
broken,  and  thereby  the  counter  E.M.F.  El  reduced  to 
zero,  and  sometimes  even  if  the  field  circuit  is  reversed 
and  the  counter  E.M.F.  El  made  negative. 

Inversely,  under  certain  conditions  of  load,  the  current 
and  the  E.M.F.  of  a  generator  do  not  disappear  if  the  gene- 
rator field  is  broken,  or  even  reversed  to  a  small  negative 
value,  in  which  latter  case  the  current  flows  against  the 
E.M.F.  EQ  of  the  generator. 

,  Furthermore,  a  shuttle  armature  without  any  winding 
will  in  an  alternating  magnetic  field  revolve  when  once 
brought  up  to  synchronism,  and  do  considerable  work  as 
a  motor. 

These  phenomena  are  not  due  to  remanant  magnetism 
nor  to  the  magnetizing  effect  of  Foucault  currents,  because 


•§§  205,  206]  REACTION  MACHINES.  309 

they  exist  also  in  machines  with  laminated  fields,  and  exist 
if  the  alternator  is  brought  up  to  synchronism  by  external 
means  and  the  remanant  magnetism  of  the  field  poles  de- 
stroyed beforehand  by  application  of  an  alternating  current. 

205.  These  phenomena  cannot  be  explained  under  the 
assumption  of  a  constant  synchronous  reactance ;  because 
in  this  case,  at  no-field  excitation,  the  E.M.F.  or  counter 
E.M.F.  of  the  machine  is  zero,  and  the  only  E.M.F.  exist- 
ing in  the  alternator  is  the  E.M.F.  of  self-induction;  that 
is,   the    E.M.F.    induced   by  the  alternating   current   upon 
itself.     If,  however,  the  synchronous  reactance  is  constant, 
the  counter  E.M.F.  of  self-induction  is  in  quadrature  with 
the  current  and  wattless;  that  is,  can  neither  produce  nor 
consume  energy. 

In  the  synchronous  motor  running  without  field  excita- 
tion, always  a  large  lag  of  the  current  behind  the  impressed 
E.M.F.  exists  ;  and  an  alternating  generator  will  yield  an 
E.M.F.  without  field  excitation,  only  when  closed  by  an 
external  circuit  of  large  negative  reactance ;  that  is,  a  circuit 
in  which  the  current  leads  the  E.M.F.,  as  a  condenser,  or 
an  over-excited  synchronous  motor,  etc. 

Self-excitation  of  the  alternator  by  armature  reaction 
can  be  explained  by  the  fact  that  the  counter  E.M.F.  of 
self-induction  is  not  wattless  or  in  quadrature  with  the  cur- 
rent, but  contains  an  energy  component  ;  that  is,  that  the 
reactance  is  of  the  form  X  =  h  — jx,  where  x  is  the  wattless 
•component  of  reactance  and  h  the  energy  component  of 
reactance,  and  h  is  positive  if  the  reactance  consumes 
power,  —  in  which  case  the  counter  E.M.F.  of  self-induc- 
tion lags  more  than  90°  behind  the  current,  —  while  h  is 
negative  if  the  reactance  produces  power,  —  in  which  case 
the  counter  E.M.F.  of  self-induction  lags  less  than  90° 
behind  the  current. 

206.  A  case  of  this  nature  has  been  discussed  already 
in  the  chapter  on  Hysteresis,  from  a  different  point  of  view. 


310  ALTERNATING-CURRENT  PHENOMENA.       [§  2O7 

There  the  effect  of  magnetic  hysteresis  was  found  to  distort 
the  current  wave  in  such  a  way  that  the  equivalent  sine 
wave,  that  is,  the  sine  wave  of  equal  effective  strength  and 
equal  power  with  the  distorted  wave,  is  in  advance  of  the 
wave  of  magnetism  by  what  is  called  the  angle  of  hysteretic 
advance  of  phase  a.  Since  the  E.M.F.  induced  by  the 
magnetism,  or  counter  E.M.F.  of  self-induction,  lags  90° 
behind  the  magnetism,  it  lags  90  +  a  behind  the  current ; 
that  is,  the  self-induction  in  a  circuit  containing  iron  is  not 
in  quadrature  with  the  current  and  thereby  wattless,  but 
lags  more  than  90°  and  thereby  consumes  power,  so  that 
the  reactance  has  to  be  represented  by  X  =  h  —jx,  where 
h  is  what  has  been  called  the  "  effective  hysteretic  resis- 
tance." 

A  similar  phenomenon  takes  place  in  alternators  of  vari- 
able reactance,  or  what  is  the  same,  variable  magnetic 
reluctance. 

207.  Obviously,  if  the  reactance  or  reluctance  is  vari- 
able, it  will  perform  a  complete  cycle  during  the  time  the 
armature  coil  moves  from  one  field  pole  to  the  next  field 
pole,  that  is,  during  one-half  wave  of  the  main  current. 
That  is,  in  other  words,  the  reluctance  and  reactance  vary 
with  twice  the  frequency  of  the  alternating  main  current. 
Such  a  case  is  shown  in  Figs.  148  and  149.  The  impressed 
E.M.F.,  and  thus  at  negligible  resistance,  the  counter  E.M.F., 
is  represented  by  the  sine  wave  E,  thus  the  magnetism  pro- 
duced thereby  is  a  sine  wave  M,  90°  ahead  of  E.  The 
reactance  is  represented  by  the  sine  wave  x,  varying  with 
tfie  double  frequency  of  E,  and  shown  in  Fig.  148  to  reach 
the  maximum  value  during  the  rise  of  magnetism,  in  Fig. 
149  during  the  decrease  of  magnetism.  The  current  /  re- 
quired to  produce  the  magnetism  <£  is  found  from  <$  and  x 
in  combination  with  the  cycle  of  molecular  magnetic  friction 
of  the  material,  and  the  power  P  is  the  product  IE  As 
seen  in  Fig.  148,  the  positive  part  of  P  is  larger  than  the 


§  2O7] 


REACTION  MACHINES. 


311 


, 


V 


V 


/\ 


-w 


\ 


74S.     Variable  Reactance,  Reaction  Machine. 


\ 


Fig.  149.     Variable  Reactance,  Reaction  Machine. 


312 


AL  TERN  A  TING-CURRENT  PHENOMENA.        [§  208 


negative  part ;  that  is,  the  machine  produces  electrical  energy 
as  generator.  In  Fig.  149  the  negative  part  of  P  is  larger 
than  the  positive ;  that  is,  the  machine  consumes  electrical 
energy  and  produces  mechanical  energy  as  synchronous 
motor.  In  Figs.  150  and  151  are  given  the  two  hysteretic 
cycles  or  looped  curves  3>,  7  under  the  two  conditions.  They 
show  that,  due  to  the  variation  of  reactance  xt  in  the  first 
case  the  hysteretic  cycle  has  been  overturned  so  as  to 
represent  not  consumption,  but  production  of  electrical 


Fig.   150.     Hysteretic  Loop  of  Reaction  Machine. 

energy,  while  in  the  second  case  the  hysteretic  cycle  has 
been  widened,  representing  not  only  the  electrical  energy 
consumed  by  molecular  magnetic  friction,  but  also  the  me- 
chanical output. 

Hence,  such  a  synchronous  motor  can  be  called  "  hyste- 
resis motor,"  since  the  mechanical  work  is  done  by  an  ex- 
tension of  the  loop  of  hysteresis. 

208.  It  is  evident  that  the  variation  of  reluctance  must 
be  symmetrical  with  regard  to  the  field  poles  ;  that  is,  that 
the  two  extreme  values  of  reluctance,  maximum  and  mini- 


2O8] 


REACTION  MACHINES. 


313 


mum,  will  take  place  at  the  moment  where  the  armature 
coil  stands  in  front  of  the  field  pole,  and  at  the  moment 
where  it  stands  midway  between  the  field  poles. 

The  effect  of  this  periodic  variation  of  reluctance  is  a 
distortion  of  the  wave  of  E.M-fF.,  or  of  the  wave  of  current, 
or  of  both.  Here  again,  as  before,  the  distorted  wave  can 
be  replaced  by  the  equivalent  sine  wave,  or  sine  wave  of 
equal  effective  intensity  and  equal  power. 

The  instantaneous  value  of  magnetism  produced  by  the 


Z 


7 


Fig.  151.     Hysteretic  Loop  of  Reaction  Machine. 

armature  current  —  which  magnetism  induces  in  the  arma- 
ture conductor  the  E.M.F.  of  self-induction  —  is  propor- 
tional to  the  instantaneous  value  of  the  current,  divided 
by  the  instantaneous  value  of  the  reluctance.  Since  the 
extreme  values  of  the  reluctance  coincide  with  the  sym- 
metrical positions  of  the  armature  with  regard  to  the  field 
poles,  —  that  is,  with  zero  and  maximum  value  of  the  in- 
duced E.M.F.,  EQ,  of  the  machine,  —  it  follows  that,  if  the 
current  is  in  phase  or  in  quadrature  with  the  E.M.F.  EQ) 
the  reluctance  wave  is  symmetrical  to  the  current  wave, 
and  the  wave  of  magnetism  therefore  symmetrical  to  the 


314  ALTERNATING-CURRENT  PHENOMENA.       [§209 

current  wave  also.  Hence  the  equivalent  sine  wave  of 
magnetism  is  of  equal  phase  with  the  current  wave  ;  that 
is,  the  E.M.F.  of  self-induction  lags  90°  behind  the  cur- 
rent, or  is  wattless. 

Thus  at  no-phase  displacement,  and  at  90°  phase  dis- 
placement, a  reaction  machine  can  neither  produce  electri- 
cal power  nor  mechanical  power. 

209.  If,  however,  the  current  wave  differs  in  phase 
from  the  wave  of  E.M.F.  by  less  than  90°,  but  more  than 
zero  degrees,  it  is  unsymmetrical  with  regard  to  the 
current  wave,  and  the  reluctance  will  be  higher  for  ris- 
ing current  than  for  decreasing  current,  or  it  will  be 
higher  for  decreasing  than  for  rising  current,  according 
to  the  phase  relation  of  current  with  regard  to  induced 
E.M.F.,  £Q. 

In  the  first  case,  if  the  reluctance  is  higher  for  rising, 
lower  for  decreasing,  current,  the  magnetism,  which  is  pro- 
portional to  current  divided  by  reluctance,  is  higher  for 
decreasing  than  for  rising  current ;  that  is,  its  equivalent 
sine  wave  lags  behind  the  sine  wave  of  current,  and  the 
E.M.F.  or  self-induction  will  lag  more  than  90°  behind  the 
current ;  that  is,  it  will  consume  electrical  power,  and 
thereby  deliver  mechanical  power,  and  do  work  as  syn- 
chronous motor. 

In  the  second  case,  if  the  reluctance  is  lower  for  rising, 
and  higher  for  decreasing,  current,  the  magnetism  is  higher 
for  rising  than  for  decreasing  current,  or  the  equivalent  sine 
wave  of  magnetism  leads  the  sine  wave  of  the  current,  and 
the  counter  E.M.F.  at  self-induction  lags  less  than  90°  be- 
hind the  current ;  that  is,  yields  electric  power  as  generator, 
and  thereby  consumes  mechanical  power. 

In  the  first  case  the  reactance  will  be  represented  by 
X  =  h  —  jx,  similar  as  in  the  case  of  hysteresis  ;  while  in 
the  second  case  the  reactance  will  be  represented  by 
X  =  -  h-  jx. 


§210]  RE  A  C  TION  MA  CHINES.  315 

210.  The  influence  of  the  periodical  variation  of  reac- 
tance will  obviously  depend  upon  the  nature  of  the  variation, 
that  is,  upon  the  shape  of  the  reactance  curve.  Since, 
however,  no  matter  what  shape  the  wave  has,  it  can  always 
be  dissolved  in  a  series  of  s&e  waves  of  double  frequency, 
and  its  higher  harmonics,  in  first  approximation  the  assump- 
tion can  be  made  that  the  reactance  or  the  reluctance  vary 
with  double  frequency  of  the  main  current  ;  that  is,  are 
represented  in  the  form, 

x  =  a  -f-  b  cos  2  <£. 

Let  the  inductance,  or  the  coefficient  of  self-induction, 
be  represented  by  — 

L  =  I  +  $  cos  2  <£» 

=  /(I  _j_  y  cos  2  <£) 

where         y  =  amplitude  of  variation  of  inductance. 

Let 

<o  =  angle  of  lag  of  zero  value  of  current  behind  maximum  value 
of  inductance  L. 

It  is  then,  assuming  the  current  as  sine  wave,  or  repla- 
cing it  by  the  equivalent  sine  wave  of  effective  intensity  /, 
•Current. 


i  =  1  VZ  sin  (p  —  u). 
The  magnetism  produced  by  this  current  is, 

^""vZf 
n 

where  n  =  number  of  turns. 
Hence,  substituted, 

sin  (ft  —  w)  (1  +  y  cos  2  ft), 


n 


•or,  expanded, 

+  £\  sin  £  cos  ft 


n 
when  neglecting  the  term  of  triple  frequency,  as  wattless. 


316  AL  TERNA  TING-CURRENT  PHENOMENA.       [§  2  1  Q 

Thus  the  E.M.F.  induced  by  this  magnetism  is, 

e=-nd-± 
dt 


hence,  expanded  — 

e  =  —  2  TT  Nil  V'2  1  (\  —  |\  cos  o>  cos  (3  +  fl  +  |\  sin  u>  sin  ft 


and  the  effective  value  of  E.M.F., 


=  2  TT  AT//4/1  -f  ^  —  y  cos  2  w. 
Hence,  the  apparent  power,  or  the  voltamperes 
y  =  IE  =  2 


E 


2-rrNl    /I    •    .I? 

V"    ~T~ 

The  instantaneous  value  of  power  is 


co  sn 


in/?  1 
and,  expanded  — 


-I  - 


sin  2  co  cos2  /?  +  sin  2  ft  I  cos  2  co  — 

Integrated,  the  effective  value  of  power  is 
P=  —  7r 


§211]  RE  A  C  TION  MA  CHINES.  317 

hence,  negative  ;  that  is,  the  machine  consumes  electrical, 
and  produces  mechanical,  power,  as  synchronous  motor,  if 
o>  >  0  ;  that  is,  with  lagging  current. 

Positive ;  that  is,  the  machine  produces  electrical,  and 
consumes  mechanical,  powe#  as  generator,  if  o>  <  0  ;  that 
is,  with  leading  current. 

The  power  factor  is 

P  y  sin  2  w 


f       P 


V+J- 

hence,  a  maximum,  if, 


y  cos  2  oi ; 


cos  2w  =  -__          8  +  y2. 


or,  expanded, 

COS  2  «  *=  -  -r  7  zsr  7 

y         4         4 

The  power,  P,  is  a  maximum  at  given  current,  /,  If 

sin  2  o>  =  1 ; 
that  is, 

£  =  45° 

at  given  E.M.F.,  E,  the  power  is 
P=  - 


f  1  -f  ^-  —  y  COS  2  w 

4 


V 

hence,  a  maximum  at 


or,  expanded, 


211.    We  have  thus,  at  impressed  E.M.F.,  E,  and  negli- 
gible resistance,  if  we  denote  the  mean  value  of  reactance, 

x=  2TrNl. 

Current  ,., 

7=  _  ^  _ 


—  y  COS  2  o>. 


318  AL  TERN  A  TING-CURRENT  PHENOMENA.       [§211 

Voltamperes, 


Power, 

£'2  y  sin  2  cu 


2^1+f-yc 

Power  factor, 

/  T-    T\  y  sin  2 

/=  cos  G£,/)  = 7=^= 


Maximum  power  at 

cos  2  w  = 

...V' 

Maximum  power  factor  at 
cos 

o>  >  0  :  synchronous  motor,  with  lagging  current, 
w  <  0  :  generator,  with  leading  current. 

As  an  instance  is  shown  in  Fig.  152,  with  angle  £  as 
abscissae,  the  values  of  current,   power,  and  power  factor, 

for  the  constants,  — 

E  =  110 
x   =  3 

y    =.8 

hence,  -  41 


p  — 


/=  cos  (£,!)  = 


Vl.45  -  cos  2  £ 

—  2017  sin  2o> 
1.45  —  cos  2  <o 

.447  sin  2  tu 


Vl.45  —  cos  2  w 


As  seen  from  Fig.  152,  the  power  factor  /  of  such  a 
machine  is  very  low^ — does  not  exceed  40  per  cent  in  this 
instance. 


§211] 


REACTION  MACHINES. 


319 


Fig.  152.    Reaction  Machine. 


320     AL  TERN  A  TING-CURRENT  PHENOMENA.    [  §§  212 ,  2 1 S 


CHAPTER    XXI. 

DISTORTION    OF    WAVE-SHAPE    AND    ITS    CAUSES. 

212.  In    the   preceding   chapters   we   have    considered 
the   alternating   currents  and   alternating   E.M.Fs.  as  sine 
waves  or  as  replaced  by  their  equivalent  sine  waves. 

While  this  is  sufficiently  exact  in  most  cases,  under 
certain  circumstances  the  deviation  of  the  wave  from  sine 
shape  becomes  of  importance,  and  no  longer,  and  it  may 
not  be  possible  to  replace  the  distorted  wave  by  an  equiv- 
alent sine  wave,  since  the  angle  of  phase  displacement 
of  the  equivalent  sine  wave  becomes  indefinite.  Thus  it 
becomes  desirable  to  investigate  the  distortion  of  the  wave, 
its  causes  and  its  effects. 

Since,  as  stated  before,  any  alternating  wave  can  be 
represented  by  a  series  of  sine  functions  of  odd  orders,  the 
investigation  of  distortion  of  wave-shape  resolves  itself  in 
the  investigation  of  the  higher  harmonics  of  the  alternating 
wave. 

In  general  we  have  to  distinguish  between  higher  har- 
monics of  E.M.F.  and  higher  harmonics  of  current.  Both 
depend  upon  each  other  in  so  far  as  with  a  sine  wave  of 
impressed  E.M.F.  a  distorting  effect  will  cause  distortion 
of  the  current  wave,  while  with  a  sine  wave  of  current 
passing  through  the  circuit,  a  distorting  effect  will  cause 
higher  harmonics  of  E.M.F. 

213.  In   a   conductor   revolving  with   uniform   velocity 
through  a  uniform  and  constant  magnetic  field,  a  sine  wave 
of  E.M.F.  is  induced.     In  a  circuit  with  constant  resistance 
and  constant  reactance,  this  sine  wave  of  E.M.F.  produces 


§214]  DISTORTION  OF   WAVE-SHAPE.  321 

a.  sine  wave  of  current.  Thus  distortion  of  the  wave-shape 
or  higher  harmonics  may  be  due  to  :  lack  of  uniformity  of 
the  velocity  of  the  revolving  conductor  ;  lack  of  uniformity 
or  pulsation  of  the  magnetic  field ;  pulsation  of  the  resis- 
tance ;  or  pulsation  of  the  'reactance. 

The  first  two  cases,  lack  of  uniformity  of  the  rotation  or 
of  the  magnetic  field,  cause  higher  harmonics  of  E.M.F.  at 
open  circuit.  The  last,  pulsation  of  resistance  and  reac- 
tance, causes  higher  harmonics  only  with  a  current  flowing 
in  the  circuit,  that  is,  under  load. 

Lack  of  uniformity  of  the  rotation  is  of  no  practical  in- 
terest as  cause  of  distortion,  since  in  alternators,  due  to 
mechanical  momentum,  the  speed  is  always  very  nearly 
uniform. 

Thus  as  causes  of  higher  harmonics  remain : 

1st.  Lack  of  uniformity  and  pulsation  of  the  magnetic 
field,  causing  a  distortion  of  the  induced  E.M.F.  at  open 
circuit  as  well  as  under  load. 

2d.  Pulsation  of  the  reactance,  causing  higher  harmonics 
under  load. 

3d.  Pulsation  of  the  resistance,  causing  higher  harmonics 
under  load  also. 

Taking  up  the  different  causes  of  higher  harmonics  we 
have  :  — 

Lack  of  Uniformity  and  Pulsation  of  tJie  Magnetic  Field. 

214.  Since  most  of  the  alternating-current  generators 
contain  definite  and  sharply  defined  field  poles  covering  in 
different  types  different  proportions  of  the  pitch,  in  general 
the  magnetic  flux  interlinked  with  the  armature  coil  will 
not  vary  as  simply  sine  wave,  of  the  form  : 

4?  cos  /?, 

but  as  a  complex  harmonic  function. 

However,  with  an  armature  of  uniform  magnetic  reluc- 
tance, in  general  the  distortion  caused  by  the  shape  of  the 


322  ALTERNATING-CURRENT  PHENOMENA.         [§214 

field  poles  is  small  and  negligible,  as  for  instance  the  curves 
Fig.  153  and  Fig.  154  show,  which  represent  the  no-load 
and  full-load  wave  of  E.M.F.  of  a  three-phase  multitooth 
alternator. 

Even  where  noticeable,  these  harmonics  can  be  consid- 
ered together  with  the  harmonics  due  to  the  varying  reluc- 
tance of  the  magnetic  circuit. 

In  ironclad  alternators  with  few  slots  and  teeth  per  pole, 
the  passage  of  slots  across  the  field  poles  causes  a  pulsation 
of  the  magnetic  reluctance,  or  its  reciprocal,  the  magnetic 
inductance  of  the  circuit.  In  consequence  thereof  the  mag- 
netism per  field  pole,  or  at  least  that  part  of  the  magnetism 
passing  through  the  armature,  will  pulsate  with  a  frequency 
2  y  if  y  =  number  of  slots  per  pole. 

Thus,  in  a  machine  with  one  slot  per  pole,  the,  instanta- 
neous magnetic  flux  interlinked  with  the  armature  con- 
ductors can  be  expressed  by  the  equation  : 

<£  =  <j>  cos  /?  {  1  +  e  cos  [2  (3  —  o>]} 

where,  <f>  =  average  magnetic  flux, 

e   =  amplitude  of  pulsation, 
and  co  =  phase  of  pulsation. 

In  a  machine  with  y  slots  per  pole,  the  instantaneous  flux 
interlinked  with  the  armature  conductors  will  be  : 


+  ecos[2r/?-  o>]}, 

if  the  assumption  is  made  that  the  pulsation  of  the  magnetic 
flux  follows  a  simple  sine  law,  as  can  be  done  approximately. 
In  general  the  instantaneous  magnetic  flux  interlinked 
with  the  armature  conductors  will  be  : 

<f>  =  3>  cos  /3  {1  +  el  cos  (2  /?  —  wj)  +  £2  cos  (4  (3  —  o>2)  +   .  .  .  }, 

where   the   terms   ev    is   predominating   if   y  =  number    of 
armature  slots  per  pole. 

This  general   equation   includes  also  the  effect  of  lack 
of  uniformity  of  the  magnetic  flux. 


§214] 


DISTORTION  OF   WAVE-SHAPE. 


323 


140 

N6  Lo 

ad 

^* 

'x, 

130 

x, 

^14 

i.5 

*=• 

2.6 

// 

S 

N 

120 

^ 

r\ 

1 

110 

^ 

i 

V 

100 

/ 

\ 

\ 

00 

i 

\ 

SO 

i 

\ 

70 

'/ 

\ 

GO 

\ 

.50 

| 

\ 

10 

/I 

\ 

30 

// 

\ 

20 

I 

V 

10 

l! 

\ 

0 
1-10 

/-- 

0 

-\ 

0 

^=- 

20 

M 

10 

V) 

5»" 
GO 

—  •  — 

70 

—  -  — 
M 

_  

90 

.•! 

100 

-=-^; 

110 

5 

•^ 

—  —  • 

140 

^^ 

150 

^^— 

1GO 

-^ 

170 

ISO 

Fig.  153.     No-load  Waue  of  E.M.F.  of  Multitooth  Three-phaser. 


130 

w 

th  L 

oad 

120 

*t 

-12 

7.0 

J%5 

3,2 

^ 

'  — 

:--- 

>x 

110 

£ 

\ 

100 

/ 

\ 

00 

, 

/ 

\x 

80 

/ 

1 

70 

/ 

t 

60 

/ 

\ 

50 

/ 

V 

40 

f    , 

f 

\ 

\ 

30 

I 

\ 

20 

1 

\ 

10 

1 

5 

0 

// 

^ 

~^. 

/-" 

—  >v 

10 

0 

10 

20 

30 

10 

50 

CO 

70 

80 

W 

100 

110 

120 

-»_ 
130 

^-  — 
140 

150 

1GO 

170 

ISO 

Fig.  154.    Full-Load  Waue  of  E.M.F.  of  Multitooth  Three-phaser. 


324  AL  TERN  A  TING-CURRENT  PHENOMENA.       [§215 

In  case  of  a  pulsation  of  the  magnetic  flux  with  the 
frequency  2y,  due  to  an  existence  of  y  slots  per  pole  in  the 
armature,  the  instantaneous  value  of  magnetism  interlinked 
with  the  armature  coil  is  : 

<£  =  3>  COS  ft  {1  +  e  COS  [2  yft  —  o>]}. 

Hence  the  E.M.F.  induced  thereby  : 

</<f> 

e  =  —  n—^- 
dt 

=  -  V2irJV«fc—  {  COS  /?  (1  +  e  COS  [2y£  -£])}. 
ff-ft 

And,  expanded  : 

e  =  V2  TT  Nn*  {sin  /?  +  e  2y~1  sin  [  (2  y  -  1)  0  -  a] 


Hence,  the  pulsation  of  the  magnetic  flux  with  the 
frequency  2  y,  as  due  to  the  existence  of  y  slots  per  pole, 
introduces  two  harmonics,  of  the  orders  (2  y  —  1)  and 
(2  y+1). 

215.    If  y=  1  it  is: 

e  =  V2  TT  Nn  3>  {sin  ft  +  1  sin  (ft  -  fi)  +  |f  sin  (3  /?  -  a)  }  ; 

^  2 

that  is  :  In  a  unitooth  single-phaser  a  pronounced  triple 
harmonic  may  be  expected,  but  no  pronounced  higher 
harmonics. 

Fig.  155  shows  the  wave  of  E.M.F.  of  the  main  coil  of 
a  standard  monocyclic  alternator  at  no  load,  represented  by  : 

e  =  E  {sin  ft  —  .242  sin  (  3  ft  —  6.3)  —  .046  sin  (5ft  —  2.6) 
+  .068  sin  (7  ft  -  3.3)  -  .027  sin'  (9  ft  -  10.0)  -  .018  sin 
(11  ft  -  6.6)  +  .029  sin  (13  ft  -  8.2)}; 

hence  giving  a  pronounced  triple  harmonic  only,  as  expected. 
If  y  =  2,  it  is  : 

e  =  V2  TT  Nn  <$>  j  sin  ft  +  —  sin  (3  ft  -  A)  +  ^  sin  (5  0  -  fi) 


§215] 


DISTORTION  OF   WAVE-SHAPE. 


325 


the  no-load  wave  of  a  unitooth  quarter-phase  machine,  hav- 
ing pronounced  triple  and  quintuple  harmonics. 
If  y  =  3,  it  is  : 


^  sin  (7  £  -  o> 

2 


That  is  :  In  a  unitooth  three-phaser,  a  pronounced  quin- 
tuple and  septuple  harmonic  may  be  expected,  but  no  pro- 
nounced triple  harmonic. 


130 


110 


100 


Sine  c 


«»=£/ 


taUsu 


cMac 


f  A  ternato 


at  ro  I 


-.027  sin(9<pr20°56>)-.018nsin  (11  ^-T 


A.M.-12-150-eOO 


ad 


2n 


S 


Co 


V    ~ 

•'n 


3L2 


-3,86 
4^! 

.72 


=  -1,4 


.   755.     No-load  Wave  of  E.M.F.  of  Unitooth  Monocyclic  Alternator. 


Fig.  156  shows  the  wave  of  E.M.F.  of  a  standard  unitooth 
three-phaser  at  no  load,  represented  by  : 

e  =  E  {sin  0  -  .12  sin  (30-  2.3)  -  .23  sin  (5  /3  -  1.5)  +  .134  sin 
(7  (3  -  6.2)  -  .002  sin  (90  +  27.7)  -  .046  sin  (11  0  - 
5.5)  +  .031  sin  (13  0  -  61.5)}. 

Thus  giving  a  pronounced  quintuple  and  septuple  and 
a  lesser  triple  harmonic,  probably  due  to  the  deviation  of 
the  field  from  uniformity,  and  deviation  of  the  pulsation 
of  reluctance  from  sine  shape. 


326  AL  TERN  A  TING-CURRENT  PHENOMENA.          [§215 


110 

130 

., 

1-20 

£ 

ne 

cor 

-po 

ien 

S          / 

^ 

f 

^ 

Co 

.  c| 

mp 

3ne 

ts 

110 

of  v 

avi 

/ 

1 

A 

/ 

Jfw 

we 

100 

f 

<P) 

=;§ 

XI 

sin 

N 

E 

^ 

/ 

'(cp 

)=d5\j 

CO 

S  IX 

_2L 

80 

70 

- 

x'c 

5  = 
if 

09,5 

2,£>] 

\ 

^ 

/ 

y*- 

0,5 

-10, 

| 

22,8 
2,4 

I 

\ 

) 

-.  05J5 
-2,95 

1 

\ 

y^ 

7,87 
^ 

60 

I, 

.5,S 

5 

jy  1 

> 

50 

§9 

\ 

y,/: 

40 

A 

nal) 

SIS  ( 

f  A 

ter 

lato 

S  V 

a  vc 

S 

\ 

so 

^u 

Tf 

(/>)• 

"ZL" 

(xi 

n  i 

•^-t- 

yi  c 

OS  i 

<7>) 

i=2 

T-1] 

\ 

20 

/ 

U 

TltO 

oth 

Thr 

ep 

last 

Ma 

chi 

le  /! 

T. 

12- 

15C 

-60 

) 

\ 

10 

/ 

Y.E.M 

F. 

at  r 

o  lo 

ad 

\ 

| 

*i** 

C, 

*^- 

32; 

-10 

0 

10 

*>*-^ 
20 

^*s* 

N 

30 

40 

50 

C 

0 

70 

E 

00 

1 

00 

110 

20 

130 

140 

150 

1GO 

170 

180 

Fig.  156.    No-load  Wave  of  E.M.F.  of  Unitooth  Three-phase  Alternator. 

In  general,  if  the  pulsation  of  the  magnetic  inductance 
is  denoted  by  the  general  expression : 


GO 

T 


the  instantaneous  magnetic  flux  is  : 


hence,  the  E.M.F. 


e= 


o.  ) 

£  ey  cos  (2  7^  —  SY)  C  ' 

1 

oo    r- 

os(^-^i)+SI|  ^  cos((27+ 


00    2y  +  l 

2 


sin  ((2 y  +  1)  £  -  £Y  )  +  €v+i  sin  ((2 


§§216,217]  DISTORTION-  OF   WAVE-SHAPE.  327 

Pulsation  of  Reactance. 

216.  The  main  causes  ©f  a  pulsation  of  reactance  are: 
magnetic  saturation  and  hysteresis,  and  synchronous  motion. 
Since  in  an  ironclad  magnetic  circuit  the  magnetism  is  not 
proportional   to  the   M.M.F.,   the  -wave  of   magnetism  and 
thus  the  wave  of  E.M.F.  will  differ  from  the  wave  of  cur- 
rent.     As  far  as  this  distortion  is  due  to  the  variation  of 
permeability,  the   distortion   is   symmetrical  and   the  wave 
of   induced  E.M.F.   represents  no   power.      The  distortion 
caused  by  hysteresis,  or  the  lag  of  the  magnetism  behind 
the  M.M.F.,  causes  an  unsymmetrical  distortion  of  the  wave 
which  makes  the  wave  of  induced   E.M.F.   differ  by  more 
than   90°  from   the   current  wave   and   thereby   represents 
power,  —  the  power  consumed  by  hysteresis. 

In  practice  both  effects  are  always  superimposed  ;  that 
is,  in  a  ferric  inductance,  a  distortion  of  wave-shape  takes 
place  due  to  the  lack  of  proportionality  between  magnetism 
and  M.M.F.  as  expressed  by  the  variation  of  the  permea- 
bility in  the  hysteretic  cycle. 

This  pulsation  of  reactance  gives  rise  to  a  distortion 
consisting  mainly  of  a  triple  harmonic.  Such  current  waves 
distorted  by  hysteresis,  with  a  sine  wave  of  impressed 
E.M.F.,  are  shown  in  Figs.  66  to  69,  Chapter  X.,  on  Hy- 
steresis. Inversely,  if  the  current  is  a  sine  wave,  the  mag- 
netism and  the  E.M.F.  will  differ  from  sine  shape. 

For  further  discussion  of  this  distortion  of  wave-shape 
by  hysteresis,  Chapter  X.  may  be  consulted. 

217.  Distortion  of  wave-shape  takes  place  also  by  the 
pulsation  of  reactance  due  to  synchronous  rotation,  as  dis- 
cussed in  chapter  on  Reaction  Machines. 

In   Figs.    148    and   149,    at   a   sine   wave    of    impressed 
E.M.F.,  the  distorted  current  waves  have  been  constructed. 
Inversely,  if  a  sine  wave  of  current, 

i  =  /cos  3 


328  A  L  TERN  A  TING-CURRENT  PHENOMENA  .        [§217 

passes  through  a  circuit  of  synchronously  varying  reac- 
tance ;  as  for  instance,  the  armature  of  a  unitooth  alterna- 
tor or  synchronous  motor  —  or,  more  general,  an  alternator 
whose  armature  reluctance  is  different  in  different  positions 
with  regard  to  the  field  poles  —  and  the  reactance  is  ex- 
pressed by 

(2  £-«)}; 


or,  more  general, 

X  =  x  \  1  +  57;  s  cos  (2  y  ft  -  3>y)  }  ; 

1 

the  wave  of  magnetism  is 


:OS  /3  —  - •<  COS  , 

^~1 

QO 


ft  -  "v)  +  ^^  cos  ((2  y  +  1)  ft  -  "y  +  1)1  1  ; 


hence  the  wave  of  induced  E.M.F. 

if  ft 


in  ((2  y  +  1)  £  -  fiv  +  1)] 


sn 


that  is,  the  pulsation  of  reactance  of  frequency,  2y,  intro- 
duces  two   higher   harmonics   of   the   order    (2  y  —  1),  and 


If    X=x{\.  +  e  cos  (2/3  —  o>)}, 


icos  (/3  -  A)  +  icos 

2  2 


^  =  ^  |  sin  ^  +  I  sin  (/3  -  S>)  +  ^  sin  (3  (3  -  co)  |  ^ 

Since  the  pulsation  of  reactance  due  to  magnetic  satu 
ration  and  hysteresis  is  essentially  of  the  frequency, 


§§218,219]       DISTORTION  OF   WAVE-SHAPE.  329 

—  that  is,  describes  a  complete  cycle  for  each  half -wave  of 
current,  —  this  shows  why  the  distortion  of  wave-shape  by 
hysteresis  consists  essentially  of  a  triple  harmonic. 

The  phase  displacement"  between  e  and  z,  and  thus  the 
power  consumed  or  produced  in  the  electric  circuit,  depend 
upon  the  angle,  o>,  as  discussed  before. 

218.  In  case  of  a  distortion  of  the  wave-shape  by 
reactance,  the  distorted  waves  can  be  replaced  by  their 
equivalent  sine  waves,  and  the  investigation  with  suffi- 
cient exactness  for  most  cases  be  carried  out  under  the 
assumption  of  sine  waves,  as  done  in  the  preceding  chapters. 

Similar  phenomena  take  place  in  circuits  containing 
polarization  cells,  leaky  condensers,  or  other  apparatus 
representing  a  synchronously  varying  negative  reactance. 
Possibly  dielectric  hysteresis  in  condensers  causes  a  dis- 
tortion similar  to  that  due  to  magnetic  hysteresis. 

Pulsation  of  Resistance. 

219.  To  a  certain  extent  the  investigation  of  the  effect 
of  synchronous  pulsation  of  the  resistance  coincides  with 
that  of  reactance ;  since  a  pulsation  of  reactance,  when 
unsymmetrical  with  regard  to  the  current  wave,  introduces 
an  energy  component  which  can  be  represented  by  an 
"effective  resistance." 

Inversely,  an  unsymmetrical  pulsation  of  the  ohmic 
resistance  introduces  a  wattless  component,  to  be  denoted 
by  "  effective  reactance." 

A  typical  case  of  a  synchronously  pulsating  resistance  is 
represented  in  the  alternating  arc. 

The  apparent  resistance  of  an  arc  depends  upon  the 
current  passing  through  the  arc  ;  that  is,  the  apparent 

resistance     Of     the     arc      =   potential  difference  between  electrodes     ^     ^     ^ 

current 

for  small  currents,  low  for  large  currents.  Thus  in  an 
alternating  arc  the  apparent  resistance  will  vary  during 


330  AL  TERNA  TING-CURRENT  PHENOMENA.        [§219 

•every  half-wave  of  current  between  a  maximum  value  at 
zero  current  and  a  minimum  value  at  maximum  current, 
thereby  describing  a  complete  cycle  per  half-wave  of  cur- 
rent. 

Let  the  effective  value  of  current  passing  through  the 
arc  be  represented  by  /. 

Then  the  instantaneous  value  of  current,  assuming  the 
current  wave  as  sine  wave,  is  represented  by 

/  =      V2  sin  ft  • 

and  the  apparent  resistance  of  the  arc,  in  first  approxima- 
tion, by 

X  =  r  (1  -f  e  cos  2  <£)  ; 

thus  the  potential  difference  at  the  arc  is 

_  ecos2  <£) 
=  rl  V2       1  -  i    sin  $  -f  1  sin  3  4  I. 


Hence  the -effective  value  of  potential  difference, 


and  the  apparent  resistance  of  the  arc, 


E 

ro  =  _  = 

The  instantaneous  power  consumed  in  the  arc  is, 
/  =  ie  =  2  r/2  j  ^1  -  -\  sin2  <#>  +  |  sin  <#>  sin  3 

Hence  the  effective  power, 


§22O]  DISTORTION  OF   WAVE-SHAPE.  331 

The  apparent  power,  or  volt  amperes  consumed  by  the 
arc,  is, 


thus  the  power  factor  of  tke  arc, 


71 

that  is,  less  than  unity. 

220.  We  find  here  a  case  of  a  circuit  in  which  the 
power  factor  —  that  is,  the  ratio  of  watts  to  volt  amperes 
—  differs  from  unity  without  any  displacement  of  phase ; 
that  is,  while  current  and  E.M.F.  are  in  phase  with  each 
other,  but  are  distorted,  the  alternating  wave  cannot  be 
replaced  by  an  equivalent  sine  wave  ;  since  the  assumption 
of  equivalent  sine  wave  would  introduce  a  phase  displace- 
ment, 

cos  co  =f 

of  an  angle,  to,  whose  sign  is  indefinite. 

As  an  instance  are  shown,  in  Fig.  157  for  the  constants, 

/=  12 
r=  3 
e  =  .9 
the  resistance, 

a  =  3  {1  +  .9  cos  2  /?)  ; 

the  current, 

/    =  17  sin  (3  • 

the  potential  difference, 

e   =  28  (sin  j3  +  .82  sin  3  (3). 

In  this  case  the  effective  E.M.F.  is 
£•=25.5; 


332  AL  TERN  A  TING-CURRENT  PHENOMENA.        [§221 

the  apparent  resistance, 


the  power, 

the  apparent  power, 

the  power  factor, 


r0  =  2.13; 

P  =  244  ; 

El  =307; 
/  =  .796. 


r\ 


ARIABLE 


=  28( 


RESISTANCE 


9  cis  2  fi) 


in/5+. 82  s 


n  3 


K 


\ 


\ 


\ 


F/g.  757.     Periodically  Varying  Resistance, 

As  seen,  with  a  sine  wave  of  current  the  E.M.F.  wave 
in  an  alternating  arc  will  become  double-peaked,  and  rise 
very  abruptly  near  the  zero  values  of  current.  Inversely, 
with  a  sine  wave  of  E.M.F.  the  current  wave  in  an  alter- 
nating arc  will  become  peaked,  and  very  flat  near  the  zero 
values  of  E.M.F. 

221.  In  reality  the  distortion  is  of  more  complex  nature  ; 
since  the  pulsation  of  resistance  in  the  arc  does- not  follow 


222] 


DISTORTION  OF   WAVE-SHAPE. 


333 


a  simple  sine  law  of  double  frequency,  but  varies  much 
more  abruptly  near  the  zero  value  of  current,  making 
thereby  the  variation  of  E.M.F.  near  the  zero  value  of 
current  much  more  abruptly,  or,  inversely,  the  variation 
of  current  more  .flat. 

A  typical  wave  of  potential  difference,  with  a  sine  wave 
of  current  passing  through  the  arc,  is  given  in  Fig.  158.* 


ONE  PAIR  CARBONS 
EGULATED  BY  HANDm 
At  Ci  dynamo   e,  m.  f,||  \ 
II.  lf  "        "      current, 
II, '       watts, 


758.     Electric  Arc, 


222.  The  value  of  e,  the  amplitude  of  the  resistance 
pulsation,  largely  depends  upon  the  nature  of  the  electrodes 
and  the  steadiness  of  the  arc,  and  with  soft  carbons  and  a 
steady  arc  is  small,  and  the  power  factor  f  of  the  arc  near 
unity.  With  hard  carbons  and  an  unsteady  arc,  e  rises 
greatly,  higher  harmonics  appear  in  the  pulsation  of  resis- 
tance, and  the  power  factor  f  falls,  being  in  extreme  cases 
even  as  low  as  .6. 

The  conclusion  to  be  drawn  herefrom  is,  that  photo- 
metric tests  of  alternating  arcs  are  of  little  value,  if,  besides 
current  and  voltage,  the  power  is  not  determined  also  by 
means  of  electro-dynamometers. 

*  From  American  Institute  of  Electrical  Engineers,  Transactions,  1890,  p. 
376.     Tobey  and  Walbridge,  on  the  Stanley  Alternate  Arc  Dynamo. 


334 


AL  TERNA  TING-CURRENT  PHENOMENA. 


223 


CHAPTER    XXII. 

EFFECTS    OF    HIGHER    HARMONICS. 

223.    To  elucidate  the  variation  in  the  shape  of  alternat- 
ing waves  caused  by  various  harmonics,  in  Figs.  159  and 


Distortion  of  Wave  Shape 

by  Triple  Harmonic 
Sin.tf-3  sin.(3/?-Gfl) 


Fig.  159.    Effect  of  Triple  Harmonic. 

160  are  shown  the  wave-forms  produced  by  the  superposi- 
tion of  the  triple  and  the  quintuple  harmonic  upon  the 
fundamental  sine  wave. 


$223]  EFFECTS   OF  HIGHER   HARMONICS.  335 

In  Fig.  159  is  shown  the  fundamental  sine  wave  and 
the  complex  waves  produced  by  the  superposition  of  a  triple 
harmonic  of  30  per  cent  the  amplitude  of  the  fundamental, 
under  the  relative  phase 'displacements  of  0°,  45°,  90°,  135°, 
and  180°,  represented  by  #ie  equations  : 

sin  ft 

sin  p  —  .3  sin  3  p 

sin  p  —  .3  sin  (3  p  —  45°) 

sin  P  -  .3  sin  (3  ft  -  90°) 

sin  p  -  .3  sin  (3  p  -  135°) 

sin  p  —  .3  sin  (3  p  —  180°). 

As  seen,  the  effect  of  the  triple  harmonic  is  in  the  first 
figure  to  flatten  the  zero  values  and  point  the  maximum 
values  of  the  wave,  giving  what  is  called  a  peaked  wave. 
With  increasing  phase  displacement  of  the  triple  harmonic, 
the  flat  zero  rises  and  gradually  changes  to  a  second  peak, 
giving  ultimately  a  flat-top  or  even  double-peaked  wave  with 
sharp  zero.  The  intermediate  positions  represent  what  is 
called  a  saw-tooth  wave. 

In  Fig.  160  are  shown  the  fundamental  sine  wave  and 
the  complex  waves  produced  by  superposition  of  a  quintuple 
harmonic  of  20  per  cent  the  amplitude  of  the  fundamental, 
under  the  relative  phase  displacement  of  0°,  45°,  90°,  135°, 
180°,  represented  by  the  equations  : 

sin  p 

sin  p  -  .2  sin  5  p 

sin  P  -  .2  sin  (5  £  -  45°) 

sin  P  —  .2  sin  (5  p  —  90°) 

sin  p  -  .2  sin  (50-  135°) 

sin  p  -  .2  sin  (5  p  —  180°). 

The  quintuple  harmonic  causes  a  flat-topped  or  even 
double-peaked  wave  with  flat  zero.  With  increasing  phase 
displacement,  the  wave  becomes  of  the  type  called,  saw- 
tooth wave  also.  The  flat  zero  rises  and  becomes  a  third 
peak,  while  of  the  two  former  peaks,  one  rises,  the  other 


336 


AL  TERN  A  TING-  CURRENT  PHENOMENA. 


[§223 


decreases,    and    the    wave    gradually   changes    to    a  triple- 
peaked  wave  with  one  main  peak,  and  a  sharp  zero. 

As  seen,  with  the  triple  harmonic,  flat-top  or  double- 
peak  coincides  with  sharp  zero,  while  the  quintuple  har- 
monic flat-top  or  double-peak  coincides  with  flat  zero. 


Distortion  of  Wave  Shape 
by  Quintuple  Harmonic 
Sin./? -.2  sin.  (5/2-55;  > 


J 


\J 


Fig.  160.     Effect  of  Quintuple  Harmonic. 


Sharp  peak  coincides  with  flat  zero  in  the  triple,  with 
sharp  zero  in  the  quintuple  harmonic. 

Thus  in  general,  from  simple  inspection  of  the  wave 
shape,  the  existence  of  these  first  harmonics  can  be  dis- 
covered. 

Some  characteristic  shapes  of  curves  are  shown  in  Fig. 
161: 


§224] 


EFFECTS   OF  HIGHER  HARMONICS. 


337 


Fig.  161.     Some  Characteristic  Wave  Shapes. 

Flat  top  with  flat  zero  : 

sin  (3  —  .15  sin  3  ft  —  .10  sin  5  p. 
Flat  top  with  sharp  zero  : 

sin  ft  -  .225  sin  (3ft  —  180°)  -  .05  sin  (5  ft  -  180°). 
Double  peak,  with  sharp  zero : 

sin  ft  -  .15  sin  (3  ft  —  180°)  —  .10  sin  5  ft. 
Sharp  peak  with  sharp  zero  : 

sin  ft  —  .15  sin  3  ft  —  .10  sin  (5  ft  —  180°). 

224.  Since  the  distortion  of  the  wave-shape  consists  in 
the  superposition  of  higher  harmonics,  that  is,  waves  of 
higher  frequency,  the  phenomena  taking  place  in  a  circuit 


338  A  L  TERN  A  TING-CURRENT  PHENOMENA .        [§225 

supplied  by  such  a  wave  will  be  the  combined  effect  of  the 
different  waves. 

Thus  in  a  non-inductive  circuit,  the  current  and  the 
potential  difference  across  the  different  parts  of  the  circuit 
are  of  the  same  shape  as  the  impressed  E.M.F.  If  self- 
induction  is  inserted  in  series  to  a  non-inductive  circuit,  the 
self-induction  consumes  more  E.M.F.  of  the  higher  harmon- 
ics, since  the  reactance  is  proportional  to  the  frequency, 
and  thus  the  current  and  the  E.M.F.  in  the  non-inductive 
part  of  the  circuit  shows  the  higher  harmonics  in  a  reduced 
amplitude.  That  is,  self-induction  in  series  to  a  non-induc- 
tive circuit  reduces  the  higher  harmonics  or  smooths  out 
the  wave  to  a  closer  resemblance  with  sine  shape.  In- 
versely, capacity  in  series  to  a  non-inductive  circuit  con- 
sumes less  E.M.F.  at  higher  than  at  lower  frequency,  and 
thus  makes  the  higher  harmonics  of  current  and  of  poten- 
tial difference  in  the  non-inductive  part  of  the  circuit  more 
pronounced  —  intensifies  the  harmonics. 

Self-induction  and  capacity  in  series  may  cause  an  in- 
crease of  voltage  due  to  complete  or  partial  resonance. 

225.  In  long-distance  transmission  over  lines  of  notice- 
able inductance  and  capacity,  rise  of  voltage  due  to  reso- 
nance may  under  circumstances  be  expected  with  higher 
harmonics,  as  waves  of  higher  frequency,  while  the  funda- 
mental wave  is"  usually  of  too  low  a  frequency  to  cause 
.  resonance. 

An  approximate  estimate  of  the  possible  rise  by  reso- 
nance with  various  harmonics  can  be  obtained  by  the  inves- 
tigation of  a  numerical  instance.  Let  in  a  long-distance 
line,  fed  by  step-up  transformers  : 

The   resistance  drop   in   the   transformers  at   full   load  =  1   per 

cent. 
The  inductance  drop  in  the   transformers  at  full  load  =  5  per 

cent  with  the  fundamental  wave. 
The  resistance  drop  in  the  line  at  full  load  — 10  per  cent. 


§  225]  EFFECTS   OF  HIGHER   HARMONICS.  339 

The  inductance  drop  in  the  line  at  full  load  =  20  per  cent  with 

the  fundamental  wave. 
The  capacity  or  charging  current  of  the  line  =20  per  cent  of  the 

full-load  current  /  at  the  frequency  of  the  fundamental. 

The  line  capacity  may  approximately  be  represented  by 
a  condenser  shunted  across  the  middle  of  the  line.  The 
E.M.F.  at  the  generator  terminals  E  is  assumed  as  main- 
tained constant. 

The  E.M.F.  consumed  by  the  resistance  of  the  circuit 
from  generator  terminals  to  condenser  is 

Ir  =  .06  E, 

or,  r  =  .06  —  . 

The  reactance  E.M.F.  between  generator  terminals  and 
condenser  is,  for  the  fundamental  frequency, 

Ix  =  .15  E, 

or,  x    =  .15  —  , 

thus  the  reactance  corresponding  to  the  frequency  (2>&  —  1) 
N  of  the  higher  harmonic  is  : 

x  (2k-  1)  =  .15  (2k  -  1)  — . 

The  capacity  current  at  fundamental  frequency  is  : 

/  =  .2  I, 
hence,  at  the  frequency  :  (2  k  —  1)  N\ 

E  ' 
if: 

e'  =  E.M.F.  of  the  (2  k  —  l)th  harmonic  at  the  condenser, 

e  =  E.M.F.  of  the  (2k  —  l)th  harmonic  at  the  generator  terminals. 

The  E.M.F.  at  the  condenser  is  :  — 

+  /.*(2£-l)2; 


340  ALTERNATING-CURRENT  PHENOMENA.         [§225 

hence,  substituted  : 


VI  -  .059856  (2  k  -  I)2  +  .0009  (2  k  -  I)4 

the  rise  of  voltage  by  inductance  and  capacity. 
Substituting : 

£=     1  2  3  4  56 

or,    2  £  -  1  =     1  3  5  7  9         11 

it  is,  a  =  1.03        1.36        3.76        2.18          .70       .38 

That  is,  the  fundamental  will  be  increased  at  open  circuit 
by  3  per  cent,  the  triple  harmonic  by  36  per  cent,  the 
quintuple  harmonic  by  276  per  cent,  the  septuple  harmonic 
by  118  per  cent,  while  the  still  higher  harmonics  are 
reduced. 

The  maximum  possible  rise  will  take  place  for : 

da 


</(2  k  -  1) 

or,  k  =  5.77. 

That  is,  at  a  frequency  : 


and  is  :  a  =  14.4. 

That  is,  complete  resonance  will  appear  at  a  frequency 
between  quintuple  and  septuple  harmonic,  and  would  raise 
the  voltage  at  this  particular  frequency  14.4  fold. 

If  the  voltage  shall  not  exceed  the  impressed  voltage  by 
more  than  100  per  cent,  even  at  coincidence  of  the  maximum 
of  the  harmonic  with  the  maximum  of  the  fundamental, 

the    triple    harmonic   must    be    less    than    70    per    cent  of    the 

fundamental, 
the  quintuple  harmonic  must  be  less  than  26.5  per  cent  of  the 

fundamental, 
the  septuple  harmonic  must  be  less  than  46  per  cent  of  the 

fundamental. 


$§226,227]          EFFECTS  OF  HIGHER  HARMONICS.  341 

The  voltage  will  not  exceed  twice  the  normal,  even  at 
a  frequency  of  complete  resonance  with  the  higher  har- 
monic, if  none  of  the  higher  harmonics  amounts  to  more 
than  7  per  cent,  of  the  fundamental. 

Herefrom  it  follows  thaf^the  danger  of  resonance  in  high 
potential  lines  is  in  general  greatly  over-estimated. 

226.  The  power  developed  by  a  complex  harmonic  wave 
in  a  non-inductive  circuit  is  the  sum  of  the  powers  of  the 
individual  harmonics.      Thus  if  upon  a  sine  wave  of  alter- 
nating E.M.F.  higher  harmonic  waves  are  superposed,  the 
effective  E.M.F.,  and  the  power  produced  by  this  wave  in  a 
given  circuit  or  with  a  given  effective  current,  are  increased. 
In  consequence  hereof  alternators  and  synchronous  motors 
•of  ironclad  unitooth  construction  —  that  is,  machines  giving 
waves  with  pronounced  higher  harmonics  —  give  with  the 
same  number  of  turns  on  the  armature,  and  the  same  mag- 
netic flux  per  field  pole  at  the  same  frequency,  a  higher 
output  than  machines  built  to  produce  sine  waves. 

227.  This  explains  an  apparent  paradox  : 

If  in  the  three-phase  star-connected  generator  with  the 
magnetic  field  constructed  as  shown  diagrammatically  in 
Fig.  162,  the  magnetic  flux  per  pole  =  M,  the  number  of 
turns  in  series  per  circuit  =  //,  the  frequency  =  N,  the 
E.M.F.  between  any  two  collector  rings  is: 


since  2;z  armature  turns  simultaneously  interlink  with  the 
magnetic  flux  3>. 

The  E.M.F.  per  armature  circuit  is  : 


hence  the  E.M.F.  between  collector  rings,  as  resultant  of 
two  E.M.Fs.  e  displaced  by  60°  from  each  other,  is  : 


342 


A L  TERNA  TING-CURRENT  PHENOMENA.        [§227 


while  the  same  E.M.F.  was  found  by  direct  calculation 
from  number  of  turns,  magnetic  flux,  and  frequency  to  be 
equal  to  2  e ;  that  is  the  two  values  found  for  the  same 
E.M.F.  have  the  proportion  V3  :  2  =  1  :  1.154. 


Fig.  162.     Three-phase  Star-connected  Alternator. 

This  discrepancy  is  due  to  the  existence  of  more  pro- 
nounced higher  harmonics  in  the  wave  e  than  in  the  wave 
E  =  e  x  V3,  which  have  been  neglected  in  the  formula  : 


e  = 


Hence  it  follows  that,  while  the  E.M.F.  between  two  col- 
lector rings  in  the  machine  shown  diagrammatically  in  Fig. 
162  is  only  e  x  V3,  by  massing  the  same  number  of  turns 
in  one  slot  instead  of  in  two  slots,  we  get  the  E.M.F.  2  e 
or  15.4  per  cent  higher  E.M.F.,  that  is,  larger  output. 


§§228,229]          EFFECTS   OF  HIGHER   HARMONICS.  343 

It  follows  herefrom  that  the  distorted  E.M.F.  wave  of 
a  unitooth  alternator  is  produced  by  lesser  magnetic  flux  per 
pole  —  that  is,  in  general,  at  a  lesser  hysteretic  loss  in  the 
armature  or  at  higher  efficiency  —  than  the  same  effective 
E.M.F.  would  be  producecP with  the  same  number  of  arma- 
ture turns  if  the  magnetic  disposition  were  such  as  to  pro- 
duce a  sine  wave. 

228.  Inversely,  if  such  a  distorted  wave   of  E.M.F.  is 
impressed  upon  a  magnetic  circuit,  as,  for  instance,  a  trans- 
former, the  wave  of  magnetism  in  the  primary  will  repeat 
in  shape  the  wave  of  magnetism  interlinked  with  the  arma- 
ture coils  of  the  alternator,  and  consequently,  with  a  lesser 
maximum  magnetic  flux,  the  same  effective  counter  E.M.F. 
will  be  produced,  that  is,  the  same  power  converted  in  the 
transformer.      Since  the  hysteretic  loss  in  the  transformer 
depends  upon  the  maximum  value  of  magnetism,  it  follows 
that  the  hysteretic  loss  in  a  transformer  is  less  with  a  dis- 
torted wave  of  a  unitooth  alternator  than  with  a  sine  wave. 

Thus  with  the  distorted  waves  of  unitooth  machines, 
generators,  transformers,  and  synchronous  motors  —  and 
induction  motors  in  so  far  as  they  are  transformers  — 
operate  more  efficiently. 

229.  From    another    side    the    same    problem    can    be 
approached. 

If  upon  a  transformer  a  sine  wave,  of  E.M.F.  is  im- 
pressed, the  wave  of  magnetism  will  be  a  sine  wave  also. 
If  now  upon  the  sine  wave  of  E.M.F.  higher  harmonics, 
as  sine  waves  of  triple,  quintuple,  etc.,  frequency  are 
superposed  in  such  a  way  that  the  corresponding  higher 
harmonic  sine  waves  of  magnetism  do  not  increase  the 
maximum  value  of  magnetism,  or  even  lower  it  by  a 
coincidence  of  their  negative  maxima  with  the  positive 
maximum  of  the  fundamental,  —  in  this  case  all  the  power 
represented  by  these  higher  harmonics  of  E.M.F.  will  be 


344      ALTERNATING-CURRENT  PHENOMENA.      [§  23O,  231 

transformed  without  an  increase  of  the  hysteretic  loss,  or 
even  with  a  decreased  hysteretic  loss. 

Obviously,  if  the  maximum  of  the  higher  harmonic  wave 
of  magnetism  coincides  with  the  maximum  of  the  funda- 
mental, and  thereby  makes  the  wave  of  magnetism  more 
pointed,  the  hysteretic  loss  will  be  increased  more  than  in 
proportion  to  the  increased  power  transformed,  i.e.,  the 
efficiency  of  the  transformer  will  be  lowered. 

That  is  :  Some  distorted  waves  of  E.M.F.  are  transformed 
at  a  lesser,  some  at  a  larger,  hysteretic  loss  than  the  sine 
wave,  if  the  same  effective  E.M.F.  is  impressed  upon  the 
transformer. 

The  unitooth  alternator  wave  belongs  to  the  former 
class  ;  the  waves  derived  from  continuous-current  machines, 
tapped  at  two  equi-distant  points  of  the  armature,  in  gen- 
eral to  the  latter  class. 

230.  Regarding  the  loss  of  energy  by  Foucault  or  eddy 
currents,   this   loss   is  not   affected   by   distortion   of  wave 
.shape,    since    the    E.M.F.    of    eddy    currents,    as    induced 
E.M.F.,    is    proportional    to    the    secondary    E.M.F.  ;    and 
thus   at   constant    impressed    primary   E.M.F.,    the   energy 
consumed   by   eddy  currents   bears   a  constant  relation   to 
the  output  of  the  secondary  circuit,  as  obvious,  since  the 
division   of  power  between   the   two   secondary  circuits  — 
the  eddy  current  circuit,  and  the  useful  or  consumer  cir- 
cuit —  is   unaffected   by   wave-shape   or  intensity   of    mag- 
netism. 

231.  In    high    potential    lines,    distorted   waves  whose 
maxima  are  very  high  above  the  effective  values,  as  peaked 
waves,  may  be   objectionable  by  increasing   the   strain   on 
the   insulation.       It   is,    however,    not    settled    yet    beyond 
doubt  whether  the  striking-distance  of   a  rapidly  alternat- 
ing potential   depends   upon   the   maximum   value   or   upon 
the    effective  value.      Since   disruptive   phenomena  do  not 


§231]  EFFECTS   OF  HIGHER  HARMONICS.  345 

always  take  place  immediately  after  application  of  the 
potential,  but  the  time  element  plays  an  important  part, 
it  is  possible  that  insulation-strain  and  striking-distance  is, 
in  a  certain  range,  dependent  upon  the  effective  potential, 
and  thus  independent  of  th&  wave-shape. 

In  general,  as  conclusions  may  be  derived  that  the  im- 
portance of  a  proper  wave-shape  is  generally  greatly  over- 
rated, but  that  in  certain  cases  sine  waves  are  desirable, 
in  other  cases  certain  distorted  waves  are  preferable. 


346    ALTERNATING-CURRENT  PHENOMENA.     [§§232,233 


CHAPTER    XXIII. 

GENERAL    POLYPHASE    SYSTEMS. 

232.  A  polyphase  system  is  an  alternating-current  sys- 
tem in  which  several  E.M.Fs.   of  the  same  frequency,  but 
displaced  in  phase  from  each  other,  produce  several  currents 
of  equal  frequency,  but  displaced  phases. 

Thus  any  polyphase  system  can  be  considered  as  con- 
sisting of  a  number,  of  single  circuits,  or  branches  of  the 
polyphase  system,  which  may  be  more  or  less  interlinked 
with  each  other. 

In  general  the  investigation  of  a  polyphase  system  is 
carried  out  by  treating  the  single-phase  branch  circuits 
independently. 

Thus  all  the  discussions  on  generators,  synchronous 
motors,  induction  motors,  etc.,  in  the  preceding  chapters, 
apply  to  single-phase  systems  as  well  as  polyphase  systems, 
in  the  latter  case  the  total  power  being  the  sum  of  the 
powers  of  the  individual  or  branch  circuits. 

If  the  polyphase  system  consists  of  n  equal  E.M.Fs. 
displaced  from  each  other  by  1  /  n  of  a  period,  the  system 
is  called  a  symmetrical  system,  otherwise  an  unsymmetrical 
system. 

Thus  the  three-phase  system,  consisting  of  three  equal 
E.M.Fs.  displaced  by  one-third  of  a  period,  is  a  symmetrical 
system.  The  quarter-phase  system,  consisting  of  two  equal 
E.M.Fs.  displaced  by  90°,  or  one-quarter  of  a  period,  is  an 
unsymmetrical  system. 

233.  The   flow  of    power   in   a  single-phase  system  is 
pulsating ;  that  is,  the  watt  curve  of  the  circuit  is  a  sine 


§233]  GENERAL   POLYPHASE   SYSTEMS.  347 

wave  of  double  frequency,  alternating  between  a  maximum 
value  and  zero,  or  a  negative  maximum  value.  In  a  poly- 
phase system  the  watt  curves  of  the  different  branches  of 
the  system  are  pulsating  also.  Their  sum,  however,  or  the 
total  flow  of  power  of  the^ystem,  may  be  either  constant 
or  pulsating.  In  the  first  case,  the  system  is  called  a 
balanced  system,  in  the  latter  case  an  unbalanced  system. 

The  three-phase  system  and  the  quarter-phase  system, 
with  equal  load  on  the  different  branches,  are  balanced  sys- 
tems ;  with  unequal  distribution  of  load  between  the  indi- 
vidual branches  both  systems  become  unbalanced  systems. 


Fig.   163. 


Fig.  164. 

The  different  branches  of  a  polyphase  system  may  be 
either  independent  from  each  other,  that  is,  without  any 
electrical  interconnection,  or  they  may  be  interlinked  with 
each  other.  In  the  first  case,  the  polyphase  system  is 
called  an  independent  system,  in  the  latter  case  an  inter- 
linked system. 

The  three-phase  system  with  star-connected  or  ring-con- 
nected generator,  as  shown  diagrammatically  in  Figs.  163 
and  164,  is  an  interlinked  system. 


348  ALTERNATING-CURRENT  PHENOMENA.         [§234 

The  four-phase  system  as  derived  by  connecting  four 
equidistant  points  of  a  continuous-current  armature  with 
four  collector  rings,  as  shown  diagrammatically  in  Fig.  165, 


~  J'E 

Fig.   165. 


is  an  interlinked  system  also.  The  four-wire  quarter-phase 
system  produced  by  a  generator  with  two  independent 
armature  coils,  or  by  two  single-phase  generators  rigidly 
connected  with  each  other  in  quadrature,  is  an  independent 
system.  As  interlinked  system,  it  is  shown  in  Fig.  166,  as 
star-connected  four-phase  system. 


—  E 


-HE 


Fig.  166. 

234.    Thus,  polyphase  systems  can  be  subdivided  into : 
Symmetrical  systems  and  unsymmetrical  systems. 
Balanced  systems  and  unbalanced  systems. 
Interlinked  systems  and  independent  systems. 
The  only  polyphase  systems  which  have  found  practical 
application  are : 

The  three-phase  system,  consisting  of  three  E.M.Fs.  dis- 


§234]  GENERAL   POLYPHASE  SYSTEMS.  349 

placed  by  one-third  of  a  period,  used  exclusively  as  inter- 
linked system. 

The  quarter-phase  system,  consisting  of  two  E.M.Fs.  in 
quadrature,  and  used  with  four  wires,  or  with  three  wires, 
which  may  be  either  an  interlinked  system  or  an  indepen- 
dent system. 


350  ALTERNATING-CURRENT  PHENOMENA.         [§235 


CHAPTER    XXIV. 

SYMMETRICAL  POLYPHASE    SYSTEMS. 

235.  If  all  the  E.M.Fs.  of  a  polyphase  system  are  equal 
in  intensity,  and  differ  from  each  other  by  the  same  angle 
of  difference  of  phase,  the  system  is  called  a  symmetrical 
polyphase  system. 

Hence,  a  symmetrical  %-phase  system  is  a  system  of  n 
E.M.Fs.  of  equal  intensity,  differing  from  each  other  in 
phase  by  I./ ;/  of  a  period : 

e1  =  E  sin 
e2  =  E  sin  (  (3 


n 

2  (n  -  1)  TT 


en  =  E  sin  (  (3  — 

n 


The  next  E.M.F.  is  again  : 

*!  =  ^  sin  08  -  2  TT)  =  ^  sin  ft. 

In  the  polar  diagram  the  n  E.M.Fs.  of  the  symmetrical 
7Z-phase  system  are  represented  by  n  equal  vectors,  follow- 
ing each  other  under  equal  angles. 

Since  in  symbolic  writing,  rotation  by  1  /  n  of  a  period, 
or  angle  2-n-/  n,  is  represented  by  multiplication  with  : 


n  n 

the  KM.Fs.  of  the  symmetrical  polyphase  system  are: 


§236]          SYMMETRICAL   POLYPHASE  SYSTEMS.  351 

20      \ 
7T       ,        .     .       Zi  7T   \  7-. 

£,  I  cos  —  .  4"  /  sin  -  \  =  j&  e  ; 
n  n 


n 


/  2  In   —    1)  7T      .        .     .       2  (»   —   1)  7T\  rr     n_l 

I  cos  —  ^  -  1  --  \-j  sin  —  ^  -  1  —  ]  =E  en  \ 


The  next  E.M.F.  is  again  : 

E  (  cos  2  TT  +/  sin  2  TT)  =  E  tn  =  E. 
Hence,  it  is 

2,T  -     .       27T  n/T 

€  =  cos +  J  sm BBS  vl. 

«  « 

Or  in  other  words  : 

In   a  symmetrical  ^-phase   system   any   E.M.F.   of   the 
system  is  expressed  by  : 


where  : 


e  = 


236.    Substituting  now  for  n  different   values,  we  get 
the  different  symmetrical  polyphase  systems,  represented  by 

/'*, 

1  n/q-  2  7T  .     .       2  7T 

where,  e  =  vl  =  cos  --  \-j  sm  -  . 

n  n 

1.)    »  =  1     c  =  1     €^  =  E, 
the  ordinary  single-phase  system. 

2.)    »  =  2    e  =  -  1     €*E=£a.nd  -  E. 


Since    —  ^  is  the  return  of  E,  n  =  2  gives  again  the 
single-phase  system. 

ON  o  2  7T  ..27T 

3.)    *  =  3     c  =  cos  -—  -  +  j  sm  —  -  = 
o  o 


352  ALTERNATING-CURRENT  PHENOMENA.        [§237 

The  three  E.M.Fs.  of  the  three-phase  system  are  : 


Consequently  the  three-phase  system  is  the  lowest  sym 
metrical  polyphase  system. 


4.)    //  =  4,    e  =  cos^-  +  /sin^-  =j,   *2=  —  1,   C3=-/. 

The  four  E.M.Fs.  of  the  four-phase  system  are  : 

<i  =  E,    JE,     -£,     -JE. 
They  are  in  pairs  opposite  to  each  other  : 

E  and  —  E  ;  j  E  and  —JE. 

Hence  can  be  produced  by  two  coils  in  quadrature  with 
each  other,  analogous  as  the  two-phase  system,  or  ordinary 
alternating-current  system,  can  be  produced  by  one  coil. 

Thus  the  symmetrical  quarter-phase  system  is  a  four- 
phase  system. 

Higher  systems,  as  the  quarter-phase  or  four-phase  sys- 
tem, have  not  been  used,  and  are  of  little  practical  interest. 

237;  A  characteristic  feature  of  the  symmetrical  n- 
phase  system  is  that  under  certain  conditions  it  can  pro- 
duce a  M.M.F.  of  constant  intensity. 

If  n  equal  magnetizing  coils  act  upon  a  point  under 
equal  angular  displacements  in  space,  and  are  excited  by  the 
n  E.M.Fs.  of  a  symmetrical  /z-phase  system,  a  M.M.F.  cf 
constant  intensity  is  produced  at  this  point,  whose  direction 
revolves  synchronously  with  uniform  velocity. 

Let, 

«'  =  number  of  turns  of  each  magnetizing  coil. 
E—  effective  value  of  impressed  E.M.F. 
/  =  effective  value  of  current. 

Hence, 
^  =///=  effective  M.M.F.  of  one  of  the  magnetizing  coils. 


§237]          SYMMETRICAL   POLYPHASE  SYSTEMS.  353 

Then  the  instantaneous  value  of  the  M.M.F.  of  the  coil 
acting  in  the  direction  2  vi  j  n  is  : 

9^.;> 


»    J 
The  two  rectangular  components  of  this  M.M.F.  are 


ZiTTt 


i'  =//cos 

n 


and  /*"  =  //  sin 

n 


Hence  the  M.M.F.  of  this  coil  can  be  expressed  by  the 
symbolic  formula  : 


fi  =  ;//V2  sin    ft  -  cos         +  /smL      . 

V  n    1  \          n  n   J 

Thus  the  total  or  resultant  M.M.F.  of  the  n  coils  dis- 
placed under  the  n  equal  angles  is  : 


1 

or,  expanded  : 


jin  £  5r     cos2-  -+/  sin  --cos 

1~  \  /z  11  n 


n  x  .""  /•     2  77-  2           z/  TT  *    .      ... 
cos  p  ?j      sin cos  +  i  sm4 

Vv     »        »  »  ; 

It  is,  however  : 

— f-  j  sin cos =  i/l-[-  cos  —  -  +  /  sin 


354  ALTERNATING-CURRENT  PHENOMENA.        [§237 

".     2iri          2?r/    .      .   .    o27rz         //H  4?r/ 

sm  -  cos  --  hysm2  -  =  £[  1  —  cos  -  -  —  /  sin 
n 

and,  since: 


n  n  n         2  «  ;/ 


-2<  =  o, 

1  1 

as  the  sum  of  all  the  roots  of   Vl, 

it  is,  /=  ;*;/(V2  (sin  /?  +  j  cos  )8). 

or,  ,.      n  nr  I   ,  ,     n    .     .         n. 

f=  —  —  -  (sm  ft  +  j  cos  (3) 

v  2 

=  ^(sin/?+/cos/3); 

V2 

the  symbolic  expression  of  the  M.M.F.  produced  by  the 
n  circuits  of  the  symmetrical  /z-phase  system,  when  exciting 
n  equal  magnetizing  coils  displaced  in  space  under  equal 
angles. 

The  absolute  value  of  this  M.M.F.  is  : 


V2    ;v5       2 

Hence  constant  and  equal  «/V2  times  the  effect've 
M.M.F.  of  each  coil  or  n/2  times  the  maximum  M.M.F. 
of  each  coil. 

The  phase  of  the  resultant  M.M.F.  at  the  time  repre- 
sented by  the  angle  (3  is  : 

tan  o>  —  cot  ft  ; 

That  is,  the  M.M.F.  produced  by  a  symmetrical  //-phase 
system  revolves  with  constant  intensity  : 

^,       n^ 

^ 

and  constant  speed,  in  synchronism  with  the  frequency  of 
the  system  ;  and,  if  the  reluctance  of  the  magnetic  circuit 


§  238]          SYMMETRICAL   POLYPHASE  SYSTEMS.  355 

is  constant,  the  magnetism  revolves  with  constant  intensity 
and  constant  speed  also,  at  the  point  acted  upon  symmetri- 
cally by  the  n  M.M.Fs.  of  the  //-phase  system. 

This  is  a  characteristic'" feature  of  the  symmetrical  poly- 
phase system.  jb 

238.  In  the  three-phase  system,  n  =  3,  F—  1.5  ^max 
where  $max  is  the  maximum  M.M.F.  of  each  of  the  magne- 
tizing coils. 

In  a  symmetrical  quarter-phase  system,  n  =  4,  F  =  2 
&max>  where  ^max  is  the  maximum  M.M.F.  of  each  of  the 
four  magnetizing  coils,  or,  if  only  two  coils  are  used,  since 
the  four-phase  M.M.F.  are  opposite  in  phase  by  two,  F '  = 
$max,  where  $max  is  the  maximum  M.M.F.  of  each  of  the 
two  magnetizing  coils  of  the  quarter-phase  system. 

While  the  quarter-phase  system,  consisting  of  two  E.M.Fs. 
displaced  by  one-quarter  of  a  period,  is  by  its  nature  an 
unsymmetrical  system,  it  shares  a  number  of  features  — 
as,  for  instance,  the  ability  of  producing  a  constant  result- 
ant M.M.F.  —  with  the  symmetrical  system,  and  may  be 
considered  as  one-half  of  a  symmetrical  four-phase  system. 

Such  systems,  of  an  even  number  of  phases,  consisting 
of  one-half  of  a  symmetrical  system,  are  called  hemisym- 
metrical  systems. 


356  ALTERNATING-CURRENT  PHENOMENA.       [§239 


CHAPTER    XXV. 

BALANCED    AND   UNBALANCED    POLYPHASE   SYSTEMS. 

239.    If  an  alternating  E.M.F.  : 

e  =•  E  V2  sin  ft 
produces  a  current  : 

i  =  7  V2  sin  OB-  w), 
where  o>  is  the  angle  of  lag,  the  power  is  : 


/  =  ei  =  2  £Ssm  ft  sin  (0  —  G) 

=  £S(cos  A  —  sin  (2  /?  —  w)), 

and  the  average  value  of  power  : 

P  =  JSfCQS  w. 

Substituting  this,  the  instantaneous  value  of  power  is 
found  as  : 

p  =  p(\.     sin  (2  ff  -a)' 

I  COS  w 

Hence  the  power,  or  the  flow  of  energy,  in  an  ordinary 
single-phase  alternating-current  circuit  is  fluctuating,  and 
varies  with  twice  the  frequency  of  E.M.F.  and  current, 
unlike  the  power  of  a  continuous-current  circuit,  which  is 

constant  : 

p  =  et. 

If  the  angle  of  lag  w  =  0  it  is  : 

/  =  P  (I  -  sin  2  ft)  • 

hence  the  flow  of  power  varies  between  zero  and  2  P,  where 
P  is  the  average  flow  of  energy  or  the  effective  power  of 
the  circuit. 


§24O]  BALANCED   POLYPHASE   SYSTEMS.  357 

If  the  current  lags  or  leads  the  E.M.F.  by  angle  £  the 
power  varies  between 


--  LA-    and 

COSw'y 


COSo> 


that  is,  becomes  "negative  for  a  certain  part  of  each  half- 
wave.  That  is,  for  a  time  during  each  half-wave,  energy 
flows  back  into  the  generator,  while  during  the  other  part 
of  the  half-wave  the  generator  sends  out  energy,  and  the 
difference  between  both  is  the  effective  power  of  the  circuit. 
If  s>  =  90°,  it  is  : 

/  =  El  cos  2/3; 

that  is,  the  effective  power  :/>  =  (),  and  the  energy  flows 
to  and  fro  between  generator  and  receiving  circuit. 

Under  any  circumstances,  however,  the  flow  of  energy  in 
the  single-phase  system  is  fluctuating  at  least  between  zero 
-and  a  maximum  value,  frequently  even  reversing. 

240.    If  in  a  polyphase  system 

eii  e2>  £3?  -   •  -  -  =  instantaneous  values  of  E.M.F.  ; 
*\>  *2j  *sj  •  •   •   •  =  instantaneous  values   of  current  pro- 
duced thereby  ; 

the  total  flow  of  power  in  the  system  is  : 


The  average  flow  of  power  is  : 

P  =  Eif-i  cos  «!  -f  £2f2  cos  w2  -f  .  .  .  . 

The  polyphase  system  is  called  a  balanced  system,  if  the 
flow  of  energy  : 

p  =  e^  +  *2/2  +  *8/8  -f  .  .  .  . 

is  constant,  and  it  is  called  an  unbalanced  system  if  the 
flow  of  energy  varies  periodically,  as  in  the  single-phase  sys- 
tem ;  and  the  ratio  of  the  minimum  value  to  the  maximum 
value  of  power  is  called  the  balance  factor  of  the  system. 


358    AL  TERN  A  TING-CURRENT  PHENOMENA.    [  §§  241 ,  242 

Hence  in  a  single-phase  system  on  non-inductive  circuit, 
that  is,  at  no-phase  displacement,  the  balance  factor  is  zero ; 
and  it  is  negative  in  a  single-phase  system  with  lagging  or 
leading  current,  and  becomes  =  —  1,  if  the  phase  displace- 
ment is  90°  —  that  is,  the  circuit  is  wattless. 

241.  Obviously,  in  a  polyphase  system  the  balance  of 
the  system  is  a  function  of  the  distribution  of  load  between 
the  different  branch  circuits. 

A  balanced  system  in  particular  is  called  a  polyphase 
system,  whose  flow  of  energy  is  constant,  if  all  the  circuits 
are  loaded  equally  with  a  load  of  the  same  character,  that 
is,  the  same  phase  displacement. 

242.  All  the  symmetrical  systems  from  the  three-phase 
system  upward  are  balanced  systems.     Many  unsymmetrical 
systems  are  balanced  systems  also. 

1.)    Three-phase  system  : 
Let 

6l  =  E  V2  sin  ft  and     t\  =  7  V2  sin  (ft  —  <o)  ; 

e2  =  E  V2  sin  (ft  -  120),  /,  =  I V2  sin  (ft  -  &  -  120)  ; 

ez  =  E  V2  sin  (ft  -  240),  /3  =  7  V2  sin  (ft  -  &  -  240)  ; 

be  the  E.M.Fs.  of  a  three-phase  system,  and  the  currents 
produced  thereby. 

Then  the  total  flow  of  power  is  : 

/  =  2  ^7 (sin  ft  sin  (ft  -  «)  +  sin  (ft  -  120)  sin  (ft  -  &  -  120) 

+  sin  08  r-  240)  sin  (0  -  o>  -  240)) 
=  3  JZfcos  ui  =  P,  or  constant. 

Hence  the  symmetrical  three-phase  system  is  a  balanced 
system. 

2.)    Quarter-phase  system  : 

Let     fl  =  E  V2  sin  ft  t\  =  7  V2  sin  (ft  -  A)  ; 

e2  =  E  V2  cosft  4  =  7  V2 cos  (ft-&)> 


§243]  BALANCED  POLYPHASE   SYSTEMS.  359 

be  the  E.M.Fs.   of  the  quarter-phase  system,  and  the  cur- 
rents produced  thereby. 

This  is  an  unsymmetrical  system,  but  the  instantaneous 
flow  of  power  is  : 

/  ==  2  Ef(sm  ft  sin  (ft  —  w)  +  cos  ft  cos  (ft  —  o>)) 
=•  2  EScos  w  =  P,  or  constant. 

Hence  the  quarter-phase  system  is  an  unsymmetrical  bal- 
anced system. 

3.)  The  symmetrical  //-phase  system,  with  equal  load 
and  equal  phase  displacement  in  all  n  branches,  is  a  bal- 
anced system.  For,  let  : 

e{  =  E  V2  sin  (  ft  -  ^  }  =  E.M.F.  ; 


/",•  =  I V2  sin  f  ft  —  w  -         -  )  =  current 
the  instantaneous  flow  of  power  is : 


1 

2  El  yTsin  (  ft  -  *±L]  sm(ft-Z-27r* 
1 


or  /  =  n  E I  =  Pt  or  constant. 

243.  An'  unbalanced  polyphase  system  is  the  so-called 
inverted  three-phase  system,  derived  from  two  branches  of 
a  three-phase  system  by  transformation  by  means  of  two 
transformers,  whose  secondaries  are  connected  in  opposite 
direction  with  respect  to  their  primaries.  Such  a  system 
takes  an  intermediate  position  between  the  Edison  three- 
wire  system  and  the  three-phase  system.  It  shares  with 
the  latter  the  polyphase  feature,  and  with  the  Edison  three- 


360 


AL  TERNA  TING-CURRENT  PHENOMENA. 


244 


wire  system  the  feature  that  the  potential  difference  be- 
tween the  outside  wires  is  higher  than  between  middle 
wire  and  outside  wire. 

By  such  a  pair  of  transformers  the  two  primary  E.M.Fs. 
of  120°  displacement  of  phase  are  transformed  into  two 
secondary  E.M.Fs.  differing  from  each  other  by  60°.  Thus 
in  the  secondary  circuit  the  difference  of  potential  between 
the  outside  wires  is  V3  times  the  difference  of  potential 
between  middle  wire  and  outside  wire.  At  equal  load  on 
the  two  branches,  the  three  currents  are  equal,  and  differ 
from  each  other  by  320°,  that  is,  have  the  same  relative 
proportion  as  in  a  three-phase  system.  If  the  load  on 
one  branch  is  maintained  constant,  while  the  load  of  the 
other  branch  is  reduced  from  equality  with  that  in  the 
first  branch  down  to  zero,  the  current  in  the  middle  wire 
first  decreases,  reaches  a  minimum  value  of  87  per  cent  of 
its  original  value,  and  then  increases  again,  reaching  at  no 
load  the  same  value  as  at  full  load. 

The  balance  factor  of  the  inverted  three-phase  system 
on  non-inductive  load  is  .333. 

244.  In  Figs.  167  to  174  are  shown  the  E.M.Fs.  as 
£  and  currents  as  i  in  drawn  lines,  and  the  power  as  /  in 
dotted  lines,  for : 


Fig.  167.     Single-phase  System  on  Non-inductiue  Load. 


§244]  BALANCED   POLYPHASE  SYSTEMS. 


361 


Fig.  168.     Single-phase  System  on  Inductive  Load  of  60°  Lag. 


"7"^ 


—  •>-*=- >-<r  — 

V'\/  \/f 


Fig.   169.     Quarter-phase  System  on  Non-inductive  Load. 


Fig.  170.     Quarter-phase  System  on  Inductive  Load  of  60°  Lag. 


362  ALTERNATING-CURRENT  PHENOMENA.       [§244 


Fig.  171.      Three-phase  System  on  Non-inductive  Load. 


Fig.  172.      Three-phase  System  on  Inductive  Load  of  60°  Lag. 


/">, 

/>V'A 
/  A  \A 


Fig.   173.     Inverted  Three-phase  System 
on  Non-inductive  Load. 


245,246]    BALANCED  POLYPHASE  SYSTEMS. 


363 


Fig.  174.    Inverted  Three-phase  System  on 
Inductive  Load  of  60°  Lag. 


245.  The  flow  of  power  in  an  alternating-current  system 
is  a  most  important  and  characteristic  feature  of  the  system, 
.and  by  its  nature  the  systems  may  be  classified  into  : 

Monocyclic  systems,  or  systems  with  a  balance  factor  zero 
or  negative. 

Polycyclic  systems,  with  a  positive  balance  factor. 

Balance  factor  —  1  corresponds  to  a  wattless  circuit, 
balance  factor  zero  to  a  non-inductive  single-phase  circuit, 
balance  factor  -+-  1  to  a  balanced  polyphase  system. 

246.  In   polar   coordinates,   the    flow   of    power   of    an 
alternating-current  system  is  represented  by  using  the  in- 
.stantaneous  flow  of  power  as  radius  vector,  with  the  angle 
,/J  corresponding   to   the  time  as   amplitude,   one  complete 
period  being  represented  by  one  revolution. 

In  this  way  the  power  of  an  alternating-current  system 
is  represented  by  a  closed  symmetrical  curve,  having  the 
;zero  point  as  quadruple  point.  In  the  monocyclic  systems 
the  zero  point  is  quadruple  nodal  point  ;  in  the  polycyclic 
system  quadruple  isolated  point. 

Thus  these  curves  are  sextics, 


364  ALTERNATING-CURRENT  PHENOMENA.      [§247 

Since  the  flow  of  power  in  any  single-phase  branch  of 
the  alternating-current  system  can  be  represented  by  a  sine 
wave  of  double  frequency  : 

sin  (2  ff  -  & 


COS 


the  total  flow  of  power  of  the  system  as  derived  by  the 
addition  of  the  powers  of  the  branch  circuits  can  be  rep- 
resented in  the  form  : 


This  is  a  wave  of  double  frequency  also,  with  c  as  ampli- 
tude of  fluctuation  of  power. 

This  is  the  equation  of  the  power  characteristics  of  the 
system  in  polar  coordinates. 

247.  To  derive  the  equation  in  rectangular  coordinates 
we  introduce  a  substitution  which  revolves  the  system  of 
coordinates  by  an  angle  w/2,  so  as  to  make  the  symmetry 
axes  of  the  power  characteristic  of  the  coordinate  axes. 

/  =  V^T7a, 

tan(0-AW. 


x 


hence,    sin  (2/3  -  *)  =  2  sin  (ft  -  f^cos  |>  -  §1  = 

\         /      L      ^  J 

substituted, 


or,  expanded  : 


the  sextic  equation  of  the  power  characteristic. 
Introducing  : 

a  =  (1  +  c)  P  =  maximum  value  of  power, 
b  =  (1  —  e)  P  =  minimum  value  of  power; 


§247]  BALANCED   POLYPHASE  SYSTEMS.  365 

it  is  P  =  ^-t—  > 


I       A  ' 
d    ~T~    0 

'      J&* 

hence,  substituted,  and  expanded : 


<*•  +  /)«  _  i  {a  ( 

the  equation   of   the  power   characteristic,   with   the   main 
power  axes  a  and  b,  and  the  balance  factor  :  b  /  a. 


It  is  thus  : 


Single-phase    non-inductive    circuit  :  p  =  P  (1  -j-   sin  2  <£), 
b  =  0,     a  =  2P— 

(*"+/)«  -  ^2  +  (*  +  j)4  =  0,      £/*  =  0.      . 

Single-phase  circuit,  60°  lag  :  p  =  P  (1  +  2  sin  2  <£),       ^  = 
-  />     tf  =  +  3  /> 


Single-phase  circuit,  90°  lag  \  p  =  E I  sin  2  <£,     b  =  —  E  Ir 
a  =  +  El  — 


Three-phase  non-inductive  circuit  :  /  =  P,     b  =  1,     a  =  1 

•*2  +/  —  ^2  =  0  :  circle,     b  /  a  =  +  1. 
Three-phase  circuit,  60°  lag  :  p  =  P,     b  =  1,     #  =  1 


Quarter-phase  non-inductive  circuit  :p  =  P,  b  =  1,    #  = 
**-+y«  -  ^2  =  0  :  circle,      bja  =  +  1. 

Quarter-phase  circuit,  60°  lag  :/=/>,     ^  =  1,     #  =  1 
•^2  +/  —  P*  =  0  :  circle.     £/a  =  +  1. 


366  ALTERNATING-CURRENT  PHENOMENA.        [§248 

Inverted  three-phase  non-inductive  circuit  : 


Inverted   three-phase   circuit  60°   lag  :/  =  P  (\.  -\-  sin  2  <£), 
=  0,     a  =  2P 


a  and  ^  are  called  the  main  power  axes  of  the  alternating- 
current  system,  and  the  ratio  b  j  a  is  the  balance  factor  of 
the  system. 


Figs.  175  and  176. 

248.  As  seen,  the  flow  of  power  of  an  alternating-cur- 
rent system  is  completely  characterized  by  its  two  main 
power  axes  a  and  b. 

The  power   characteristics  in   polar  coordinates,    corre- 


§  248] 


BALANCED   POLYPHASE   SYSTEM. 


867 


spending  to  the  Figs.  167,  168,  173,  and  174  are  shown  in 
Figs.  175,  176,  177,  and  178. 


Figs.  177  and  178. 


The  balanced  quarter-phase  and  three-phase  systems  give 
as  polar  characteristics  concentric  circles. 


368    ALTERNATING-CURRENT  PHENOMENA.     [§§249,250 


CHAPTER    XXVI. 

POLYPHASE    SYSTEMS. 


249.  In  a  polyphase   system   the  different  circuits  of 
displaced  phases,  which  constitute  the  system,  may  either 
be  entirely  separate  and  without  electrical  connection  with 
each   other,   or   they   may   be   connected   with    each   other 
electrically,  so  that  a  part  of  the  electrical  conductors  are 
in   common  to  the  different  phases,  and  in  this  case  the 
system  is  called  an  interlinked  polyphase  system. 

Thus,  for  instance,  the  quarter-phase  system  will  be 
called  an  independent  system  if  the  two  E.M.Fs.  in  quadra- 
ture with  each  other  are  produced  by  two  entirely  separate 
coils  of  the  same,  or  different  but  rigidly  connected,  arma- 
tures, and  are  connected  to  four  wires  which  energize  inde- 
pendent circuits  in  motors  or  other  receiving  devices.  If 
the  quarter-phase  system  is  derived  by  connecting  four 
equidistant  points  of  a  closed-circuit  drum  or  ring-wound 
armature  to  the  four  collector  rings,  the  system  is  an  inter- 
linked quarter-phase  system. 

Similarly  in  a  three-phase  system.  Since  each  of  the 
three  currents  which  differ  from  each  other  by  one-third 
of  a  period  is  equal  to  the  resultant  of  the  other  two  cur- 
rents, it  can  be  considered  as  the  return  circuit  of  the  other 
two  currents,  and  an  interlinked  three-phase  system  thus 
consists  of  three  wires  conveying  currents  differing  by  one- 
third  of  a  period  from  each  other,  so  that  each  of  the  three 
currents  .is  a  common  return  of  the  other  two,  and  inversely. 

250.  In  an  interlinked  polyphase  system  two  ways  exist 
of  connecting  apparatus  into  the  system. 


§250]  INTERLINKED  POLYPHASE  SYSTEMS. 


369 


1st.  The  star  connection,  represented  diagram  mat  ically 
in  Fig.  179.  In  this  connection  the  n  circuits  excited  by 
currents  differ  from  each  other  by  1  /  n  of  a  period,  and  are 
connected  with  their  one*  end  together  into  a  neutral  point 
or  common  connection,  wftich  may  either  be  grounded  or 
connected  with  other  corresponding  neutral  points,  or  insu- 
lated. 

In  a  three-phase  system  this  connection  is  usually  called 
a  Y  connection,  from  a  similarity  of  its  diagrammatical  rep- 
resentation with  the  letter  Y,  as  shown  in  Fig.  163. 


2d.  The  ring  connection,  represented  diagrammatically 
in  Fig.  180,  where  the  n  circuits  of  the  apparatus  are  con- 
nected with  each  other  in  closed  circuit,  and  the  corners 
or  points  of  connection  of  adjacent  circuits  connected  to 
the  n  lines  of  the  polyphase  system.  In  a  three-phase 
system  this  connection  is  called  the  delta  connection,  from 
the  similarity  of  its  diagrammatic  representation  with  the 
Greek  letter  Delta,  as  shown  in  Fig.  164.  • 

In  consequence  hereof  we  distinguish  between  star- 
connected  and  ring-connected  generators,  motors,  etc.,  or 


370  ALTERNATING-CURRENT  PHENOMENA.        [§251 

...  6 


Fig.   180. 

in    three-phase  systems   Y- connected    and    delta-connected 
apparatus. 

251.  Obviously,  the  polyphase  system  as  a  whole  does 
not  differ,  whether  star  connection  or  ring  connection  is 
used  in  the  generators  or  other  apparatus ;  and  the  trans- 
mission line  of  a  symmetrical  //-phase  system  always  con- 
sists of  n  wires  carrying  current  of  equal  strength,  when 
balanced,  differing  from  each  other  in  phase  by  l/;z  of  a 
period.  Since  the  line  wires  radiate  from  the  n  terminals 
of  the  generator,  the  lines  can  be  considered  as  being  in 
star  connection. 

The  circuits  of  all  the  apparatus,  generators,  motors, 
etc.,  can  either  be  connected  in  star  connection,  that  is, 
between  one  line  and  a  neutral  point,  or  in  ring  connection, 
Jhat  is,  between  two  lines. 

In  general  some  of  the  apparatus  will  be  arranged  in 
star  connection,  some  in  ring  connection,  as  the  occasion 
may  require. 


§§252,253]      INTERLINKED  POLYPHASE  SYSTEMS.       371 

252.  In  the  same  way  as  we  speak  of  star  connection 
and  ring  connection  of  the  circuits  of   the  apparatus,  the 
term  star  potential  and  ring  potential,  star  current  and  ring 
current,  etc.,  are  used,  ,whereby  as  star   potential  or  in  a 
three-phase  circuit  Y  poteij£ial,  the  potential  difference  be- 
tween one  of  the  lines  and  the  neutral  point,  that  is,  a  point 
having  the  same  difference  of  potential  against  all  the  lines, 
is  understood  ;  that  is,  the  potential  as  measured  by  a  volt- 
meter  connected   into   star  or  Y  connection.      By   ring   or 
delta  potential    is   understood    the  difference   of    potential 
between   adjacent   lines,  as  measured  by  a  voltmeter  con- 
nected  between   adjacent  lines,    in   ring   or  delta   connec- 
tion. 

In  the  same  way  the  star  or  Y  current  is  the  current 
flowing  from  one  line  to  a  neutral  .point  ;  the  ring  or  delta 
current,  the  current  flowing  from  one  line  to  the  other. 

.  The  current  in  the  transmission  line  is  always  the  star 
or  Y  current,  and  the  potential  difference  between  the  line 
wires,  the  ring  or  delta  potential. 

Since  the  star  potential  and  the  ring  potential  differ 
from  each  other,  apparatus  requiring  different  voltages  can 
be  connected  into  the  same  polyphase  mains,  by  using  either 
star  or  ring  connection.  The  total  power  of  the  polyphase 
system  is  equal  to  the  sum  of  all  the  star  or  Y  powers,  or 
to  the  sum  of  all  the  ring  or  delta  powers. 

253.  If  in  a  generator  with  star-connected  circuits,  the 
E.M.F.    per   circuit  =  E,  and   the   common   connection   or 
neutral  point  is  denoted  by  zero,  the  potentials  of  the  n 
terminals  are  : 


or  in  general  :  e*  E, 

at  the  zth  terminal,  where  : 

*  =  0,  1,  2.  .  .  .  n-  1,     e  =  cos—  +/sin—  =  VI 


372  ALTERNATING-CURRENT  PHENOMENA.        [§253 

Hence  the  E.M.F.  in  the  circuit  from  the  zth  to  the  £th 
terminal  is  : 

Eki  =  JE  -  ?E  =  (e*  -  e«)  E. 

The  E.M.F.  between  adjacent  terminals  i  and  i  +  1  is  : 

(e«'  +  i  _  €t)  E  =  e«  (e  -  1)  £. 

In  a  generator  with  ring-connected  circuits,  the  E.M.F. 
per  circuit  : 

is  the  ring  E.M.F.,  and  takes  the  place  of 


while  the  E.M.F.  between  terminal   and  neutral  point,  or 
the  star  E.M.F.,  is  : 

iff* 

Hence  in  a  star-connected  generator  with  the  E.M.F. 
E  per  circuit,  it  is  : 

Star    E.M.F.,  j  E. 

Ring  E.M.F.,  e*  (e  -  1)  E. 

E.M.F.  between  terminal  i  and  terminal  k  (c*  —  e*)  E. 

In  a  ring-connected  generator  with  the  E.M.F.  E  per 
circuit,  it  is  : 

Star    E.M.F.,  -^—  E. 
e  —  1 

Ring  E.M.F.,  c*  E. 

,lc  _  fi 

E.M.F.  between  terminals  i  and  /£,  -     —  E. 

e  —  -  1 

In  a  star-connected  apparatus,  the  E.M.F.  and  the  cur- 
rent per  circuit  have  to  be  the  star  E.M.F.  and  the  star 
current.  In  a  ring-connected  apparatus  the  E.M.F.  and 
current  per  circuit  have  to  be  the  ring  E.M.F.  and  ring 
current. 

In  the  generator  of  a  symmetrical  polyphase  system,  if  : 
e*  E  are  the  E.M.Fs.  between  the  n  terminals  and  the 
neutral  point,  or  star  E.M.Fs., 


-§  254]          INTERLINKED   POLYPHASE  SYSTEMS.  373 

/,-  =  the  currents  issuing  from  terminal  i  over  a  line  of 
the  impedance  Z{  (including  generator  impedance  in  star 
connection),  we  have  : 

Potential  at  end  of  line  i  : 

JE  -  Z^. 

Difference  of  potential  between  terminals  k  and  i  : 
(€*-c*)£-  (Zklk  -Z,/,), 

where  72-  is  the  star  current  of  the  system,  Z{  the  star  im- 
pedance. 

The  ring  potential  at  the  end  of  the  line  between  ter- 
minals i  and  k  is  Eikt  and  it  is  : 

Eik  =  -Eki. 

If  now  Iik  denotes  the  current  passing  from  terminal  i  to 
terminal  k,  and  Zik  impedance  of  the  circuit  between  ter- 
minal i  and  terminal  k,  where  : 

fit  =  ~  /«, 

Zik  =  Zki, 
it  is  Eik  =  ZikIik. 

If  Iio  denotes  the  current  passing  from  terminal  i  to  a 
ground  or  neutral  point,  and  Zio  is  the  impedance  of  this 
circuit  between  terminal  i  and  neutral  point,  it  is  : 


254.    We  have  thus,  by  Ohm's  law  and  KirchhorFs  law  : 

If  €*  E  is  the  E.M.F.  per  circuit  of  the  generator,  be- 
tween the  terminal  i  and  the  neutral  point  of  the  generator, 
or  the  star  E.M.F. 

I{  =  the  current  issuing  from  the  terminal  i  of  the  gen- 
erator, or  the  star  current. 

Z{  =  the  impedance  of  the  line  connected  to  a  terminal 
i  of  the  generator,  including  generator  impedance. 

Ei  =  the  E.M.F.  at  the  end  of  line  connected  to  a  ter- 
minal i  of  the  generator. 


374  ALTERNATING-CURRENT  PHENOMENA.        [§254 

Eik  =  the  difference  of  potential  between  the  ends  of 
the  lines  i  and  k. 

Iik  =  the  current  passing  from  line  i  to  line  k. 

Zik  =  the  impedance  of  the  circuit  between  lines  i  and  k. 

Iio>  two  -  -  •  -  =  the  current  passing  from  line's"  to  neu- 
tral points  0,  00,  .  .  .,  . 

Zio,  Zioo  .  .  .  .  =  the  impedance  of  the  circuits  between 
line  i  and  neutral  points  0,  00,  .... 

It  is  then  : 


1.)    Eilc  =  —  £MJ     fit  =  —  fid,     Zilc  =  ZM,     Iio  =  —  /ot> 

Zio  =  Zoi,  etc. 
2.)    Ei    =  SE-Zifi. 
3.)    Ei    =  Ziolio  =  Zioolioo  =  .  .  .  . 
4.)    Eik  =  Ek-  Ei  =  (e*  -  e<)  E  -  (Zklk  -  Z«/«). 

5-)     -Eik.~ 

6.)     I,    = 

0 

7.)  If  the  neutral  point  of  the  generator  does  not  exist, 
as  in  ring  connection,  or  is  insulated  from  the  other  neutral 
points  : 


=  0  ; 
=«0,  etc, 


Where  0,  00,  etc.,  are  the  different  neutral  points  which 
are  insulated  from  each  other. 

If  the  neutral  point  of  the  generator  and  all  the  other 
neutral  points  are  grounded  or  connected  with  each  other, 
it  is  : 


§254]  INTERLINKED   POLYPHASE   SYSTEMS.  375. 

If  the  neutral  point  of  the  generator  and  all  other  neu- 
tral points  are  grounded,  the  system  is  called  a  grounded 
system.  If  the  neutral  points  are  not  grounded,  the  sys- 
tem is  an  insulated  polyphase  system,  and  an  insulated 
polyphase  system  -with  equalizing  return,  if  all  the  neutral 
points  are  connected  with  each  other. 

8.)    The  power  of  the  polyphase  system  is  — 


:*  Eli  cos  fa  at  the  generator 
1 

n          n 

=  y*    y*  Eik  Iik  cos  <j>ijk  in  the  receiving  circuits. 

0        i 


376    ALTERNATING-CURRENT  PHENOMENA.    [§§255,256 


CHAPTER    XXVII. 

TRANSFORMATION    OF   POLYPHASE   SYSTEMS. 

255.  In  transforming  a  polyphase  system  into  another 
polyphase  system,   it  is   obvious  that  the  primary  system 
must  have  the  same  flow  of  power  as  the  secondary  system, 
neglecting  losses  in  transformation,  and  that  consequently 
a  balanced  system  will  be  transformed  again  in  a  balanced 
system,  and  an  unbalanced  system  into  an  unbalanced  sys- 
tem of  the  same  balance  factor,  since  the  transformer  is  an 
apparatus  not  able  to  store  energy,  and  thereby  to  change 
the   nature  of  the  flow  of  power.      The  energy  stored  as 
magnetism,  amounts  in  a  well-designed  transformer  only  to 
a  very  small  percentage  of  the  total  energy.      This  shows 
the  futility  of    producing  symmetrical  balanced   polyphase 
systems  by  transformation  from  the  unbalanced  single-phase 
system  without    additional   apparatus  able  to  store  energy 
efficiently,  as   revolving  machinery. 

Since  any  E.M.F.  can  be  resolved  into,  or  produced  by, 
two  components  of  given  directions,  the  E.M.Fs.  of  any 
polyphase  system  can  be  resolved  into  components  or  pro- 
duced from  components  of  two  given  directions.  This  en- 
ables the  transformation  of  any  polyphase  system  into  any 
other  polyphase  system  of  the  same  balance  factor  by  two 
transformers  only. 

256.  Let   Elt    Ez,    Ez  .  .  .  .  be   the  E.M.Fs.   of  the 
primary  system  which  shall  be  transformed  into  — 

EI,  EI,  ^  ....  the  E.M.Fs.  of  the  secondary 
system. 

Choosing  two   magnetic   fluxes,    </>   and   ^,    of   different 


§  256]    TRANSFORMATION  OF  POLYPHASE  SYSTEMS.     377 

phases,  as  magnetic  circuits  of  the  two  transformers,  which 
induce  the  E.M.Fs.,  e  and  F,  per  turn,  by  the  law  of  paral- 
lelogram the  E.M.Fs.,  El,  E2,  ....  can  be  dissolved  into 
two  components,  El  and  E^ ,  E2  and  E2 ,  .  .  .  .  of  the  phases, 
e  and  J. 
Then,  - 

£1,  £2,  .  .  .  .  are  the  counter  E.M.Fs.  which  have  to  be 
induced  in  the  primary  polyphase  circuits  of  the  first 
transformer ; 

hence 

EI  I  e,  E% 1  ~e  .  .  .  .  are  the  numbers  of  turns  of  the  primary 
coils  of  the  first  transformer. 

Analogously 

EI  IT     Ez  /J .  .  .  .  are  the  number  of  turns  of  the  primary  coils 
in  the  second  transformer. 

In  the  same  manner  as  the  E.M.Fs.  of  the  primary 
system  have  been  resolved  into  components  in  phase  with 
e  and  e,  the  E.M.Fs.  of  the  secondary  system,  E^9  Ez\  .... 

are  produced  from  components,  E^  and  E^,  E^-  and  E^1, 
....  in  phase  with  e  and  7,  and  give  as  numbers  of  second- 
ary turns,— 

EI  /  <?,  E2l  /  ^,  ....  in  the  first  transformer  ; 
E±  1 7,  £2l  /  F,  ....  in  the  second  transformer. 

That  means  each  of  the  two  transformers  m  and  m  con- 
tains in  general  primary  turns  of  each  of  the  primary 
phases,  and  secondary  turns  of  each  of  the  secondary 
phases.  Loading  now  the  secondary  polyphase  system  in 
any  desired  manner,  corresponding  to  the  secondary  cur- 
rents, primary  currents  will  flow  in  such  a  manner  that  the 
total  flow  of  power  in  the  primary  polyphase  system  is  the 
same  as  the  total  flow  of  power  in  the  secondary  system, 
plus  the  loss  of  power  in  the  transformers. 


378  ALTERNATING-CURRENT  PHENOMENA.        [§257 

257.  As  an  instance  may  be  considered  the  transforma- 
tion of  the  symmetrical  balanced  three-phase  system 

E  sin  ft     E  sin  (ft  —  120),      E  sin  (0  —  240), 
in  an  unsymmetrical  balanced  quarter-phase  system : 

4 

E'  sin  ft     E'  sin  (ft  —  90). 
Let  the  magnetic  flux  of  the  two  transformers  be 

<f>  cos  ft   and    </>  cos  (ft  —  90). 

Then  the  E.M.Fs.  induced  per  turn  in  the  transformers 
are  £  sin  ft  and  e  sin  (/3  -  90)  ; 

hence,  in  the  primary  circuit  the  first  phase,  E  sin  ft  will 
give,  in  the  first  transformer,  Ej e  primary  turns;  in  the 
second  transformer,  0  primary  turns. 

The  second  phase,  E  sin  (ft  —  120),  will  give,  in  the 
first  transformer,  —  E  /  2  e  primary  turns;  in  the  second 

transformer,  — 2-    -  primary  turns. 

The  third  phase,  E  sin  (ft  —  240),  will  give,  in  the  first 
transformer,  —  E  J  %  e  primary  turns;  in  the  second  trans- 

-  E  xV3 

former,  primary  turns. 

2  e 

In  the  secondary  circuit  the  first  phase  E'  sin  ft  will  give 
in  the  first  transformer:  E' I  e  secondary  turns;  in  the 
second  transformer  :  0  secondary  turns. 

The  second  phase  :  E'  sin  (ft  —  90)  will  give  in  the  first 
transformer  :  0  secondary  turns ;  in  the  second  transformer, 
E' I  e  secondary  turns. 

Or,  if : 

E  =  5,000     E'  =  100,     e  =  10. 

PRIMARY.  SECONDARY. 

1st.  2d.  3d.  1st.       2d.        Phase. 

first  transformer  +  500       -  250      -  250     10       0 

second  transformer  0     +  433     —  433       0     10     turns. 


§  258]     TRANSFORMATION  OF  POLYPHASE  SYSTEMS.    379 

That  means  :    „ 

Any  balanced  polyphase  system  can  be  transformed  by  two 
transformers  only,  without,  storage  of  energy,  into  any  other 
balanced  polyphase  system.  { 

258.  Transformation  with  a  change  of  the  balance 
factor  of  the  system  is  possible  only  by  means  of  apparatus 
able  to  store  energy,  since  the  difference  of  power  between 
primary  and  secondary  circuit  has  to  be  stored  at  the  time 
when  the  secondary  power  is  below  the  primary,  and  re- 
turned during  the  time  when  the  primary  power  is  below 
the  secondary.  The  most  efficient  storing  device  of  electric 
energy  is  mechanical  momentum  in  revolving  machinery. 
It  has,  however,  the  disadvantage  of  requiring  attendance. 


380  ALTERNATING-CURRENT  PHENOMENA.       [§259 


CHAPTER    XXVIII. 

COPPER  EFFICIENCY  OF  SYSTEMS. 

259.  In  electric  power  transmission  and  distribution^ 
wherever  the  place  of  consumption  of  the  electric  energy 
is  distant  from  the  place  of  production,  the  conductors 
which  transfer  the  current  are  a  sufficiently  large  item  to 
require  consideration,  when  deciding  which  system  and 
what  potential  is  to  be  used. 

In  general,  in  transmitting  a  given  amount  of  power  at  a 
given  loss  over  a  given  distance,  other  things  being  equal, 
the  amount  of  copper  required  in  the  conductors  is  inversely 
proportional  to  the  square  of  the  potential  used.  Since 
the  total  power  transmitted  is  proportional  to  the  product 
of  current  and  E.M.F.,  at  a  given  power,  the  current  will 
vary  inversely  proportional  to  the  E.M.F.,  and  therefore,, 
since  the  loss  is  proportional  to  the  product  of  current- 
square  and  resistance,  to  give  the  same  loss  the  resistance 
must  vary  inversely  proportional  to  the  square  of  the  cur~ 
rent,  that  is,  proportional  to  the  square  of  the  E.M.F.  ;  and 
since  the  amount  of  copper  is  inversely  proportional  to  the 
resistance,  other  things  being  equal,  the  amount  of  copper 
varies  inversely  proportional  to  the  square  of  the  E.M.F, 
used. 

This  holds  for  any  system. 

Comparing  now  the  different  systems,  as  two-wire 
single-phase,  single-phase  three-wire,  three-phase  and  quar- 
ter-phase, as  basis  of  comparison  equality  of  the  potential 
is  used. 

Some  systems,  however,  as  for  instance,  the  Edison- 
three-wire  system,  or  the  inverted  three-phase  system,  have- 


§  26OJ  COPPER   EFFICIENCY  OF  SYSTEMS.  381 

different  potentials  in  the  different  circuits  constituting  the 
system,  and  thus  the  comparison  can  be  made  either  — 

1st.  On  the  basis  of  equality  of  the  maximum  potential 
difference  in  the  system  *  or 

2d.  On  the  ba^sis  of  tJie  minimum  potential  difference 
in  the  system,  or  the  potential  difference  per  circuit  or 
phase  of  the  system. 

In  low  potential  circuits,  as  secondary  networks,  where 
the  potential  is  not  limited  by  the  insulation  strain,  but  by 
the  potential  of  the  apparatus  connected  into  the  system, 
as  incandescent  lamps,  the  proper  basis  of  comparison  is 
equality  of  the  potential  per  branch  of  the  system,  or  per 
phase. 

On  the  other  hand,  in  long  distance  transmissions  where 
the  potential  is  not  restricted  by  any  consideration  of  ap- 
paratus suitable  for  a  certain  maximum  potential  only,  but 
where  the  limitation  of  potential  depends  upon  the  problem 
of  insulating  the  conductors  against  disruptive  discharge, 
the  proper  comparison  is  on  the  basis  of  equality  of  the 
maximum  difference  of  potential  in  the  system ;  that  is, 
equal  maximum  dielectric  strain  on  the  insulation. 

The  same  consideration  holds  in  moderate  potential 
power  circuits,  in  considering  the  danger  to  life  from  wires 
or  high  differences  of  potential 

Thus  the  comparison  of  different  systems  of  long-dis- 
tance transmission  at  high  potential  or  power  distribution 
for  motors  is  to  be  made  on  the  basis  of  equality  of  the 
maximum  difference  of  potential  existing  in  the  system. 
The  comparison  of  low  potential  distribution  circuits  for 
lighting  on  the  basis  of  equality  of  the  minimum  difference 
of  potential  between  any  pair  of  wires  connected  to  the 
receiving  apparatus. 

260.  1st.  Comparison  on  the  basis  of  equality  of  the 
minimum  difference  of  potential,  in  low  potential  lighting 
circuits : 


382  ALTERNATING-CURRENT  PHENOMENA.        [§  26O 

In  the  single-phase  alternating-current  circuit,  if  e  = 
E.M.F.,  i=  current,  r=  resistance  per  line,  the  total  power 
is  =  ei,  the  loss  of  power  2z'2r. 

Using,  however,  a  three-wire  system,  the  potential  be- 
tween outside  wires  and  neutral  being  given  =  e,  the 
potential  between  the  outside  wires  is  =  2  e,  that  is,  the  dis- 
tribution takes  place  at  twice  the  potential,  or  only  £  the 
copper  is  needed  to  transmit  the  same  power  at  the  same 
loss,  if,  as  it  is  theoretically  possible,  the  neutral  wire  has  ' 
no  cross-section.  If  therefore  the  neutral  wire  is  made  of 
the  same  cross-section  with  each,  of  the  outside  wires,  f  of 
the  copper  of  the  single-phase  system  is  needed ;  if  the 
neutral  wire  is  ^  the  cross-section  of  each  of  the  outside 
wires,  ^\  of  the  copper  is  needed.  Obviously,  a  single- 
phase  five-wire  system  will  be  a  system  of  distribution  at 
the  potential  4  e,  and  therefore  require  only  TV  of  the  copper 
of  the  single-phase  system  in  the  outside  wires  ;  and  if  each 
of  the  three  neutral  wires  is  of  ^  the  cross-section  of  the 
outside  wires,  /T  ==  10.93  per  cent  of  the  copper  of  a  single- 
phase  system. 

Coming  now  to  the  three-phase  system  with  the  poten- 
tial e  between  the  lines  as  delta  potential,  if  i  =  the  current 
per  line  or  Y  current,  the  current  from  line  to  line  or  delta 
current  =  ^  /  V3  ;  and  since  three  branches  are  used,  the 
total  power  is  3  e  ^  /  V3  =  e  i^  VS.  Hence  if  the  same 
power  has  to  be  transmitted  by  the  three-phase  system  as 
with  the  single-phase  system,  the  three-phase  line  current 
must  be  ^  =  i  j  VS  ;  hence  if  r^  =  resistance  of  each  of  the 
three  wires,  the  loss  per  wire  is  if  r^  =  z'Vj/3,  and  the 
total  loss  is  i 2  r\ ,  while  in  the  single-phase  system  it  is 
2  i\.  Hence,  to  get  the  same  loss,  it  must  be :  r^  =  rt  that 
is,  each  of  the  three  three-phase  lines  has  twice  the  resis- 
tance —  that  is,  half  the  copper  of  each  of  the  two  single- 
phase  lines ;  or  in  other  words,  the  three-phase  system 
requires  three-fourths  of  the  copper  of  the  single-phase 
.system  of  the  same  potential. 


§  26O]  COPPER  EFFICIENCY  OF  SYSTEMS.  383 

Introducing,  however,  a  fourth  neutral  wire  into  the 
three-phase  system,  and  connecting  the  lamps  between  the 
neutral  wire  and  the  three  outside  wires  —  that  is,  in  Y  con- 
nection —  the  potential  between  the  outside  wires  or  delta 
potential  will  be  =•=  e  X  V5,  since  the  Y  potential  =  e,  and 
the  potential  of  the  system  is  raised  thereby  from  e  to 
e  V3  ;  that  is,  only  \  as  much  copper  is  required  in  the  out- 
side wires  as  before  —  that  is  \  as  much  copper  as  in  the 
single-phase  two-wire  system.  Making  the  neutral  of  the 
same  cross -section  as  the  outside  wires,  requires  \  more 
copper,  or  £  =  33.3  per  cent  of  the  copper  of  the  single- 
phase  system  ;  making  the  neutral  of  half  cross-section, 
requires  \  more,  or  ^7¥  =  29.17  per  cent  of  the  copper  of 
the  single-phase  system.  The  system,  however,  now  is  a 
four-wire  system. 

The  independent  quarter-phase  system  with  four  wires 
is  identical  in  efficiency  to  the  two-wire  single-phase  sys- 
tem, since  it  is  nothing  but  two  independent  single-phase 
systems  in  quadrature. 

The  four-wire  quarter-phase  system  can  be  used  as  two 
independent  Edison  three-wire  systems  also,  deriving  there- 
from the  same  saving  by  doubling  the  potential  between 
the  outside  wires,  and  has  in  this  case  the  advantage,  that 
by  interlinkage,  the  same  neutral  wire  can  be  used  for  both 
phases,  and  thus  one  of  the  neutral  wires  saved. 

In  this  case  the  quarter-phase  system  with  common  neu- 
tral of  full  cross-section  requires  T5g  =  31.25  per  cent,  the 
quarter-phase  system  with  common  neutral  of  one-half  cross- 
section  requires  ^  =  28.125  per  cent,  of  the  copper  of  the 
two-wire  single-phase  system. 

In  this  case,  however,  the  system  is  a  five-wire  system, 
and  as  such  far  inferior  to  the  five-wire  single-phase  system. 

Coming  now  to  the  quarter-phase  system  with  common 
return  and  potential  e  per  branch,  denoting  the' current  in 
the  outside  wires  by  /2,  the  current  in  the  central  wire  is 
22  V2  ;  and  if  the  same  current  density  is  chosen  for  all 


384  ALTERNATING-CURRENT  PHENOMENA.        [§  26Q 

three  wires,  as  the  condition  of  maximum  efficiency,  and 
the  resistance  of  each  outside  wire  denoted  by  r2,  the  re- 
sistance of  the  central  wire  =  rz/V2,  anc*  the  loss  of  power 
per  outside  wire  is  if  r2 ,  in  the  central  wire  2  z'22  r2  /  V2 
=  ifr^  V2;  hence  the  total  loss  of  power  is  2  z'22r2  +  z'22;2 
V2  =  z'22  r2  (2  -f  V2).  The  power  transmitted  per  branch 
is  z'2  e,  hence  the  total  power  2  z'2  e.  To  transmit  the  same 
power  as  by  a  single-phase  system  of  power,  eiy  it  must 

be  z2  =7/2;  hence  the  loss,  *V2<2  +  ^) .  Since  this 
loss  shall  be  the  same  as  the  loss  2z'2r  in  the  single- 
phase  system,  it  must  be  2  r  =  — 


,  or  r~  = 


fcv 

r 


2  +  V2 

9     _L      A/^ 

Therefore  each  of  the  outside  wires  must  be  -  -  times 

as  large  as  each  single-phase  wire,  the  central  wire  V2 
times  larger ;  hence  the  copper  required  for  the  quarter- 
phase  system  with  common  return  bears  to  the  copper 
required  for  the  single-phase  system  the  relation  : 


>)  | 

per  cent  of  the  copper  of  the  single-phase  system. 

Hence  the  quarter-phase  system  with  common  return 
saves  2  per  cent  more  copper  than  the  three-phase  system, 
but  is  inferior  to  the  single-phase  three-wire  system. 

The  inverted  three-phase  system,  consisting  of  two 
E.M.Fs.  e  at  60°  displacement,  and  three  equal  currents 
ts  in  the  three  lines  of  equal  resistance  r3,  gives  the  out- 
put 2<?z3,  that  is,  compared  with  the  single-phase  system, 
z'3  =  z'/ 2.  The  loss  in  the  three  lines  is  3  z32  ;3  =  f  z'2  r3. 
Hence,  to  give  the  same  loss  2  z2  r  as  the  single-phase  sys- 
tem, it  must  be  r3  =  f  r,  that  is,  each  of  the  three  wires 
must  have  f  of  the  copper  cross-section  of  the  wire  in  the 
two-wire  single-phase  system  ;  or  in  other  words,  the  in- 
verted three-phase  system  requires  T\  of  the  copper  of  the 
two-wire  single-phase  system. 


§  260] 


COPPER   EFFICIENCY  OF  SYSTEMS. 


385 


We  get  thus  the  result, 

If  a  given  power  has.  to  be  transmitted  at  a  given  loss, 
and  a  given  minitmim  potential,  as  for  instance  110  volts 
for  lighting,  the  amount  of  copper  necessary  is  : 

2  WIRES  :    Single-phase  system,  ;    .  ;        100.0 

3  WIRES  :    Edison    three-wire    single-phase    sys- 

tem,  neutral  full  section,  37.5 
Edison    three-wire    single-phase    sys- 
tem, neutral  half-section,  31.25 
Inverted  three-phase  system,  56.25 
Quarter-phase  system  with  common 

return,  72.9 


Three-phase  system, 
4  WIRES  :    Three-phase,  with   neutral  wire,  full 


ro.O 


section,  33.3 

Three-phase,  with  neutral  wire,  half- 
section,  29.17 
Independent  quarter-phase  system,      100.0 

5  WIRES  :  Edison  five-wire,  single-phase  system, 

full  neutral,  15.625 

Edison  five-wire,  single-phase  system, 

half-neutral,  10.93 

Four-wire,  quarter-phase,  with  com- 
mon neutral,  full  section,  31.25 

Four-wire,  quarter-phase,  with  com- 
mon neutral,  half-section,  28.125 

We  see  herefrom,  that  in  distribution  for  lighting  —  that 
is,  with  the  same  minimum  potential,  and  with  the  same 
number  of  wires  —  the  single-phase  system  is  superior  to 
any  polyphase  system. 

The  continuous-current  system  is  equivalent  in  this 
comparison  to  the  single-phase  alternating-current  system 
of  the  same  effective  potential,  since  the  comparison  is 
made  on  the  basis  of  effective  potential,  and  the  power 
depends  upon  the  effective  potential  also. 


386  ALTERNATING-CURRENT  PHENOMENA.        [§261 

261.  Comparison  on  the  Basis  of  Equality  of  the  Maximum 
Difference  of  Potential  in  the  System,  in  Long-Distance 
Transmission,  Power  Distribution,  etc. 

Wherever  the  potential  is  so  high  as  to  bring  the  ques- 
tion of  the  strain  on  the  insulation  into  consideration,  or  in 
other  cases,  to  approach  the  danger  limit  to  life,  the  proper 
comparison  of  different  systems  is  on  the  basis  of  equality 
of  maximum  potential  in  the  system. 

Hence  in  this  case,  since  the  maximum  potential  is 
fixed,  nothing  is  gained  by  three-  or  five-wire  Edison  sys- 
tems. Thus,  such  systems  do  not  come  into  consideration. 

The  comparison  of  the  three-phase  system  with  the 
•single-phase  system  remains  the  same,  since  the  three- 
phase  system  has  the  same  maximum  as  minimum  poten- 
tial ;  that  is  : 

The  three-phase  system  requires  three-fourths  of  the 
copper  of  the  single-phase  system  to  transmit  the  same 
power  at  the  same  loss  over  the  same  distance. 

The  four-wire  quarter-phase  system  requires  the  same 
amount  of  copper  as  the  single-phase  system,  since  it  con- 
sists of  two  single-phase  systems. 

In  a  quarter-phase  system  with  common  return,  the 
potential  between  the  outside  wire  is  V2  times  the  poten- 
tial per  branch,  hence  to  get  the  same  maximum  strain  on 
the  insulation  —  that  is,  the  same  potential  e  between  the 
outside  wires  as  in  the  single-phase  system  —  the  potential 
per  branch  will  be  el  V2,  hence  the  current  z'4  =  i  j  V2,  if  i 
equals  the  current  of  the  single-phase  system  of  equal 
power,  and  z"4  V2  =  i  will  be  the  current  in  the  central 
wire. 

Hence,  if  r4  =  resistance  per  outside  wire,  r4  /  V2  = 
resistance  of  central  wire,  and  the  total  loss  in  the  sys- 
tem is  : 

2  ,-v4  +         L  =  ,;«,,  (2  +  V2)  = ,-  v,  <£±L . 


§  261  ]  COPPER   EFFICIENCY  OF  SYSTEMS.  387 

Since  in  the  single-phase  system,  the  loss  =  2  z2  r,  it  is  : 


.2+  V2 

'  2  4-  V2  • 

That  is,  each  of  the  outride  wires  has  to  contain  -  —  — 

times  as  much  copper  as  each  of   the   single-phase  wires. 

2  x  V2    /- 
The    central    wires    have    to    contain  --  V  2   times  as 

2  (2  -f-  V2) 
much  copper  ;  hence  the  total  system  contains  - 

2  +V2     _ 
—  r  -  V2  times  as  much  copper  as  each  of  the  single- 

3-1-2  A/2 
phase  wires  ;    that  is,  -  -  times  the  copper  of  the 

single-phase  system. 
Or,  in  other  words, 
A  quarter-phase  system  with  common  return  requires 

3-4-2  V2 

-  =  1.45T  times  as  much  copper  as  a  single-phase 

system  of  the  same  maximum  potential,  same  power,  and 
same  loss. 

Since  the  comparison  is  made  on  the  basis  of  equal 
maximum  x  potential,  and  the  maximum  potential  of  alter-: 
nating  system  is  V2  times  that  of  a  continuous-current 
circuit  of  equal  effective  potential,  the  alternating  circuit 
of  effective  potential  e  compares  with  the  continuous-cur- 
rent circuit  of  potential  e  V2,  which  latter  requires  only 
half  the  copper  of  the  alternating  system. 

This  comparison  of  the  alternating  with  the  continuous- 
current  system  is  not  proper  however,  since  the  continuous- 
current  potential  introduces,  besides  the  electrostatic  strain, 
an  electrolytic  strain  on  the  dielectric  which  does  not  exist 
in  the  alternating  system,  and  thus  makes  the  action  of  the 
continuous-current  potential  on  the  insulation  more  severe 
than  that  of  an  equal  alternating  potential.  Besides,  at  the 
voltages  which  came  under  consideration,  the  continuous 
current  is  excluded  to  begin  with. 


388  ALTERNATING-CURRENT  PHENOMENA.        [§262 

Thus  we  get : 

If  a  given  power  is  to  be  transmitted  at  a  given  loss, 
and  a  given  maximum  difference  of  potential  in  the  system, 
that  is,  with  the  same  strain  on  the  insulation,  the  amount 
of  copper  required  is  : 

2  WIRES  :  Single-phase  system,  100.0 

Continuous-current  system,  50.0 

3  WIRES  :  Three-phase  system,  75.0 

Quarter-phase  system,  with  common  return,  145.7 

4  WIRES  :  Independent  Quarter-phase  system,  100.0 

Hence  the  quarter-phase  system  with  common  return  is 
practically  excluded  from  long-distance  transmission. 

262.  In  a  different  way  the  same  comparative  results 
between  single-phase,  three-phase,  and  quarter-phase  sys- 
tems can  be  derived  by  resolving  the  systems  into  their 
single-phase  branches. 

The  three-phase  system  of  E.M.F.  e  between  the  lines 
can  be  considered  as  consisting  of  three  single-phase  cir- 
cuits of  E.M.F.  */V3,  and  no  return.  The  single-phase 
system  of  E.M.F.  e  between  lines  as  consisting  of  two 
single-phase  circuits  of  E.M.F.  e/2  and  no  return.  Thus, 
the  relative  amount  of  copper  in  the  two  systems  being 
inversely  proportional  to  the  square  of  E.M.F.,  bears  the 
relation  ( v3  /  e)2  :  (2  /  e)2  =  3  :  4  ;  that  is,  the  three-phase 
system  requires  75  per  cent  of  the  copper  of  the  single- 
phase  system. 

The  quarter-phase  system  with  four  equal  wires  requires 
the  same  copper  as  the  single-phase  system,  since  it  consists 
of  two  single-phase  circuits.  Replacing  two  of  the  four 
quarter-phase  wires  by  one  wire  of  the  same  cross-section 
as  each  of  the  wires  replaced  thereby,  the  current  in  this 
wire  is  V2  times  as  large  as  in  the  other  wires,  hence,  the 
loss  twice  as  large  —  that  is,  the  same  as  in  the  two  wires 
replaced  by  this  common  wire,  or  the  total  loss  is  not 


§  262]  COPPER   EFFICIENCY  OF  SYSTEMS.  389 

changed  —  while  25  per  cent  of  the  copper  is  saved,  and 
the  system  requires  only  75  per  cent  of  the  copper  of  the 
single-phase  system,  but  produces  V2  times  as  high  a 
potential  between  the  outside  wires.  Hence,  to  give  the 
same  maximum  potential,  'The  E.M.Fs.  of  the  system  have 
to  be  reduced  by  V2,  that  is,  the  amount  of  copper  doubled, 
and  thus  the  quarter-phase  system  with  common  return  of 
the  same  cross-section  as  the  outside  wires  requires  150 
per  cent  of  the  copper  of  the  single-phase  system.  In  this 
case,  however,  the  current  density  in  the  middle  wire  is 
higher,  thus  the  copper  not  used  most  economical,  and 
transferring  a  part  of  the  copper  from  the  outside  wire  to 
the  middle  wire,  to  bring  all  three  wires  to  the  same  current 
density,  reduces  the  loss,  and  thereby  reduces  the  amount 
of  copper  at  a  given  loss,  to  145.7  per  cent  of  that  of  a 
single-phase  system. 


390  ALTERNATING-CURRENT  PHENOMENA.         [§263 


CHAPTER    XXIX. 

THREE-PHASE    SYSTEM. 

263.  With  equal  load  of  the  same  phase  displacement 
in  all  three  branches,  the  symmetrical  three-phase  system 
offers  no  special  features  over  those  of  three  equally  loaded 
single-phase  systems,  and  can  be  treated  as  such  ;  since  the 
mutual  reactions  between  the  three-phases  balance  at  equal 
distribution  of  load,  that  is,  since  each  phase  is  acted  upon 
by  the  preceding  phase  in  an  equal  but  opposite  manner 
as  by  the  following  phase. 

With  unequal  distribution  of  load  between  the  different 
branches,  the  voltages  and  phase  differences  become  more  or 
less  unequal.  These  unbalancing  effects  are  obviously  maxi- 
mum, if  some  of  the  phases  are  fully  loaded,  others  unloaded,. 

Let: 

E  =  E.M.F.  between  branches  1  and  2  of  a  three-phaser, 
Then: 

c  E  =  E.M.F.  between  2  and  3, 
=  E.M.F.  between  3  and  1, 


where,  «  =  ^i=  ~        -. 

Let 

Zi9  Z<t,  Zz  =  impedances  of  the  lines  issuing  from  genera- 

tor terminals  1,  2,  3, 
and   Y19  K2,  Ys  =  admittances   of   the   consumer  circuits   con- 

nected between  lines  2  and  3,  3  and  1,  1  and  2. 
If  then, 

fi,  /2,  /8,  are  the  currents  issuing  from  the  generator  termi- 
nals into  the  lines,  it  is, 

/!+/,  +  /,  =  0.  (1) 


§263]  THREE-PHASE   SYSTEM.  391 

If     //,  /2?  //=  currents  flowing  through  the  admittances    Yl9, 
F2,  K8,  from  2  to  3,  3  to  1,  1  to  2,  it  is, 

A  =  /.'  -  //,  >  or,   I,  +  72'  -  /,'  =  O 

(2) 


These  three  equations  (2)  added,  give  (1)  as  dependent 
equation. 

At  the  ends  of  the  lines  1,  2,  3,  it  is  : 


£V  =  ^2_Z3/3  +  Z1/  (3) 

^/  =  ^3_Zi/i  +  Z2/2J 

the  differences  of  potential,  and 

^ 

(4) 


the  currents  in  the  receiver  circuits. 

These  nine  equations   (2),   (3),   (4),  determine  the   nine 
quantities  :  715  72,  73,  7/,  72',  73',  £"/,  ^2',  ^3'. 

Equations  (4)  substituted  in  (2)  give  : 


(5) 


These  equations  (5)  substituted  in   (3),  and  transposed, 
give, 

since      El  =  e  E 

£2  =  £  E     as  E.M.Fs.  at  the  generator  terminals. 


e  E  -  E^  (1  +  FiZg  +  r^)  +  ^/  F2Z8  +  ^  1",  Z2  =  0] 

#E  -  E,f  (1  +  F2Z8  +  r.ZO  +  ^/  FaZ!  +  ^/  Fj  Z3  =  0  I      (6) 

£•  -  ^/  (i  +  YSZ,  +  r3z2)  +  ^/  F^  +  ^/  F2  zx  =  o  I 


392  AL TERNA  TING-CURRENT  PHENOMENA.        [§263 

as  three    linear    equations  with    the    three  quantities  £\', 
F '    F ' 

J-^n   )    •*-'&  • 

Substituting  the  abbreviations  : 


-  (1  +  FjZ,  +  PiZ8),      F2Z3, 
P1Z8,       -(1  +  FaZa  +  PiZ,), 
^,      KsZ!,      -  (1  +   F3Z!  +  Pg 


FZ 


.,       F2Z1?      - 


-(1  +  F3Z1  +  F3Z2)  , 
Z2  -f-  FiZg),       F2Z3?      e 


= 


it  is: 


(8) 


D 


//  = 


-/      Y.  A 


hence, 


+/2    -f/a 


=  0 
=0 


(10) 


(11) 


§  264]  THREE-PHASE  SYSTEM. 

264.    SPECIAL    CASES. 

A.    Balanced  System 


Substituting  this  in  (6),  and  transposing : 

*E 


3FZ 


3FZ 


£*'=* 


1  +  3KZ 


EY 


1  +  3FZ 


f  3FZ 
EY 

f  3  FZJ 


1  +  3FZ 

_  (c-l)^F 
1  4-3FZ 


1+3FZ 


(12) 


The  equations  of  the  symmetrical  balanced  three-phase 
system. 

B.    One  circuit  loaded,  two  unloaded: 


Substituted  in  equations  (6)  : 
e  E  -  E{  +  E{  FZ  =  0 


unloaded  branches. 


E  -  Ezr(l  +  2  FZ)  =  0,  loaded  branch. 


hence  : 


2FZ 


1  +  2FZ 


1  +  2 


unloaded ; 


loaded  ; 


all  three 

E.M.FS.      (13) 

unequal. 


394 


AL  TERN  A  TING-CURRENT  PHENOMENA .        [§264 


•*» 

Si 

4 

=    /2 

';;   ] 

1 

+  2FZ      J 
EY           \ 

1 

+  2  YZ 
EY 

=  0 

1  +  2FZ 

(13) 


(13) 


C.    Two  circuits  loaded,  one  unloaded, 

Y1  =  Y2  =  Y,      Y3  =  0, 
Z,  =  Z2  =  Z3  =  Z. 

Substituting  this  in  equations  (6),  it  is  : 

e^  -  E{  (1  +  2  YZ)  +  EJYZ  =  01  . 

*E  -  El  (1  +  2  YZ)  +  E{  YZ  =  o}  loaded  branches' 

E  -  E{  +  (E{  +  ^/)  FZ  =  0     unloaded  branch. 
or,  since  : 


E  -  E    - 


E'  = 


thus: 


1  +YZ 


4  FZ  +  3  F2Z2 


loaded  branches. 


unloaded  branch. 


-      (W) 


As   seen,    with   unsymmetrical  distribution  of  load,   all 
three  branches  become  more  or  less  unequal. 


§  265]  QUARTER-PHASE   SYSTEM.  395 


CHAPTER    XXX. 

QTJAKTER-PHASE    SYSTEM. 

265.  In  a  three-wire  quarter-phase  system,  or  quarter- 
phase  system  with  common  return  wire  of  both  phases,  let 
the  two  outside  terminals  and 'wires  be  denoted  by  1  and  2, 
the  middle  wire  or  common  return  by  0. 

It  is  then  : 

EI  =  E  =  E.M.F.  between  0  and  1  in  the  generator. 
£2  =  j  E  =  E.M.F.  between  0  and  2  in  the  generator. 

Let: 

7j  and  72  —  currents  in  1  and  in  2, 
f0  =  current  in  0, 

Zl  and  Z2  =  impedances  of  lines  1  and  2, 
Z0  =  impedance  of  line  0. 

Yl  and  Y2  =  admittances  of  circuits  0  to  1,  and  0  to  2, 
//  and  72'=  currents  in  circuits  0  to  1,  and  0  to  2, 
E^ndE^  =  potential  differences  at  circuit  0  to   1,  and 
Oto2. 

it  is  then,  7:  +  7a  +  70  =  0  )  ~, 

or,  70  =  -  (7,  +  72)  ) 

that  is,  f0  is  common  return  of  I±  and  72. 

Further,  let : 


£*'  =JE  -  72  Z0  +  70Z0  =JE  -  72  (Z2  +  Z0)  - 
and 


(3) 


396  ALTERNATING-CURRENT  PHENOMENA.        [§266 

Substituting  (3)  in  (2)  ;  and  expanding  : 
'  =        _  1  +  F2Z2  +  F2Z0(1  -y) 


F2z0  +  F2z2)  -  K.F^ 


(1  +  F^  +  PiZ,)  (1  +  F2  Z0  +  F2  Z2)  -  F!  F2  Z02 

Hence,  the  two  E.M.Fs.  at  the  end  of  the  line  are  un- 
equal in  magnitude,  and  not  in  quadrature  any  more. 

266.    SPECIAL  CASES  : 

A.    Balanced  System. 

Z]  =  Z2  =  Z; 

Z0  =  Z/V2; 
Y,  =  F2  =  F 

Substituting  these  values  in  (4),  gives  : 


=    E 


1  +  V2  (1  +  V2)  YZ  +  (1  +  V2)  F2Z2 

1  +  (1.707  -  .707/)  FZ 
1  +  3.414  FZ  +  2.414  F2Z2 


(5), 


1  +  V2  (1  +  V2)  FZ  +  (1  +  V2)  F2Z: 

1  +  (1.7Q7  +  .7Q7y)  YZ 
1  +  3.414  FZ  +  2.414  F2Z2 

Hence,  the  balanced  quarter-phase  system  with  common 
return  is  unbalanced  with  regard  to  voltage  and  phase  rela- 
tion, or  in  other  words,  even  if  in  a  quarter-phase  system  with 
common  return  both  branches  or  phases  are  loaded  equally, 
with  a  load  of  the  same-  phase  displacement,  nevertheless 
the  system  becomes  unbalanced,  and  the  two  E.M.Fs.  at 
the  end  of  the  line  are  neither  equal  in  magnitude,  nor  in 
quadrature  with  each  other. 


§266]  QUARTER-PHASE  SYSTEM. 

B.    One  branch  loaded,  one  unloaded. 


397 


Y,  =  j),      F2  =  F 

Fx  =  F,      F2  =  0. 

Substituting  these  values  in  (4),  gives  : 
1  +  FZ1  +  V^-y 

.£1  =  v£    •• ~ "•"" ~~ ^^~ 


*•) 


V2 


FZ 


V2 


V2 


1  - 


f~JJB 


=  E\          2.414  +  1^1 
FZ 
1 


1-Fz 


=y^ 


1  +  V2 
V2 
1 


1.707  FZ 


?/  =  ^ 


1  +  FZ 


1  + V2 


/=y^ 


1  +  1.707  FZ 

+  FZi  + V2+y 

V2 

1  +  FZM^V2 

V2 


FZ 


1  + 


=  ^^  2.414 -ft 


FZ 


(6) 


398  ALTERNATING-CURRENT  PHENOMENA.        [§266 

These  two  E.M.Fs.  are  unequal,  and  not  in  quadrature 
with  each  other. 

But  the  values  in  case  a.}  are  different  from  the  values 
in  case  b.). 

That  means  : 

The  two  phases  of  a  three-wire  quarter-phase  system 
are  unsymmetrical,  and  the  leading  phase  1  reacts  upon 
the  lagging  phase  2  in  a  different  manner  than  2  reacts 
upon  1. 

It  is  thus  undesirable  to  use  a  three-wire  quarter-phase 
system,  except  in  cases  where  the  line  impedances  Z  are 
negligible. 

In  all  other  cases,  the  four-wire  quarter-phase  system 
is  preferable,  which  essentially  consists  of  two  independent 
single-phase  circuits,  and  is  treated  as  such. 

Obviously,  even  in  such  an  independent  quarter-phase 
system,  at  unequal  distribution  of  load,  unbalancing  effects 
may  take  place. 

If  one  of  the  branches  or  phases  is  loaded  differently 
from  the  other,  the  drop  of  voltage  and  the  shift  of  the 
phase  will  be  different  from  that  in  the  other  branch  ;  and 
thus  the  E.M.Fs.  at  the  end  of  the  lines  will  be  neither 
equal  in  magnitude,  nor  in  quadrature  with  each  other. 

With  both  branches  however  loaded  equally,  the  system 
remains  balanced  in  voltage  and  phase,  just  like  the  three- 
phase' system  under  the  same  conditions. 

Thus  the  four-wire  quarter-phase  system  and  the  three- 
phase  system  are  balanced  with  regard  to  voltage  and  phase 
at  equal  distribution  of  load,  but  are  liable  to  become  un- 
balanced at  unequal  distribution  of  load ;  the  three-wire 
quarter-phase  system  is  unbalanced  in  voltage  and  phase, 
even  at  equal  distribution  of  load. 


APPENDICES, 


APPENDIX    I 


ALGEBRA   OF   COMPLEX    IMAGINARY 
QUANTITIES. 

INTRODUCTION. 

267.  The  system   of   numbers,   of  which   the    science 
of   algebra  treats,   finds   its   ultimate  origin  in  experience. 
Directly  derived  .  from    experience,  however,  are   only  the 
absolute  integral  numbers  ;  fractions,  for  instance,  are  not 
directly  derived  from  experience,  but  are  abstractions  ex- 
pressing relations  between  different  classes  of  quantities. 
Thus,  for  instance,  if  a  quantity  is  divided  in  two  parts, 
from  one  quantity  two  quantities  are  derived,  and  denoting 
these  latter  as  halves  expresses  a  relation,  namely,  that  two 
of  the  new  kinds  of  quantities  are  derived  from,  or  can  be 
combined  to  one  of  the  old  quantities. 

268.  Directly  derived  from  experience  is  the  operation 
of  counting  or  of  numeration. 

a,     a  +  1,     a  +  2,     a  +  3   .  ... 
Counting  by  a  given  number  of  integers  : 


b  integers 
introduces  the  operation  of  addition^  as  multiple  counting  : 

a  +  b  =  c. 

It  is,  a  +  b  =  b  +  a, 

401 


402  APPENDIX  I.  [§269 

that  is,  the  terms  of  addition,  or  addenda,  are  interchange- 
able. 

Multiple  addition  of  the  same  terms  : 

a  -f  a  +  a  +  •  •  •  +  a  =  c 

b  equal  numbers 
introduces  the  operation  of  multiplication  : 

a  X  b  =  e. 
It  is,  a  X  b  =  b  X  a, 

that  is,  the  terms  of  multiplication,  or  factors,  are   inter- 
changeable. 

Multiple  multiplication  of  the  same  factors  : 

a  X  a  X  a  X  .   .   .  X  a  =  c 


b  equal  numbers 
introduces  the  operation  of  involution  : 

ab  =  c 
Since  ab  is  not  equal  to  ba, 

the  terms  of  involution  are  not  interchangeable. 

269.  The  reverse  operation  of  addition  introduces  the 
operation  of  subtraction : 

If  a  +  b  =  c, 

it  is  c  —  b  =  a. 

This  operation  cannot  be  carried  out  in  the  system  of 

absolute  numbers,  if : 

b>  c. 

Thus,  to  make  it  possible  to  carry  out  the  operation  of 
subtraction  under  any  circumstances,  the  system  of  abso- 
lute numbers  has  to  be  expanded  by  the  introduction  of 

the  negative  number: 

-  a  =  (-  1)  X  a, 

where  (—  1) 

is  the  negative  unit. 

Thereby  the  system  of  numbers   is   subdivided   in   the 


§§270,271]      COMPLEX  IMAGIATARY  QUANTITIES.  403 

positive  and  negative  numbers,  and  the  operation  of  sub- 
traction possible  for  all  values  of  subtrahend  and  minuend. 

or  (-  1,)  -X  (-  1)  =  1  ; 

that  is,"  the  negative  unit  TS  defined  by  : 


270.  The  reverse  operation  of  multiplication  introduces 
the  operation  of  division: 

If  a  X  b  =  c. 

then  -—a. 

b 

In  the  system  of  integral  numbers  this  operation  can  only 
be  carried  out,  if  b  is  a  factor  of  c. 

To  make  it  possible  to  carry  out  the  operation  of  division 
under  any  circumstances,  the  system  of  integral  numbers 
has  to  be  expanded  by  the  introduction  of  the  fraction  : 


where 

b 

is  the  integer  fraction,  and  is  denned  by  : 


271.    The  reverse  operation  of  involution  introduces  two 
new  operations,  since  in  the  involution : 

#*=c, 
the  quantities  a  and  b  are  not  reversible. 

Thus  ^  =  a,  the  evolution. 

loga  c  =  b,  the  logarithmation. 

The   operation  of  evolution  of  terms  c,  which  are  not 
complete  powers,  makes  a  further  expansion  of  the  system 


404  APPENDIX  I.  [§§272,  273 

of  numbers  necessary,  by  the  introduction  of  the  irrational 
number  (endless  decimal  fraction),  as  for  instance  : 

V2  =  1.414213. 

272.    The  operation  of  evolution  of  negative  quantities 
c  with  even  exponents  b,  as  for  instance 


makes  a  further  expansion  of  the  system  of  numbers  neces- 
sary, by  the  introduction  of  the  imaginary  unit. 

Thus 


where  :    V  —  1  is  denoted  by  /. 

Thus,  the  imaginary  unity  is  defined  by  : 

y2  =  -  1. 

By  addition  and  subtraction  of  real  and  imaginary  units, 
compound  numbers  are  derived  of  the  form  : 


which  are  denoted  as  complex  imaginary  numbers. 

No  further  system  of  numbers  is  introduced  by  the 
operation  of  evolution. 

The  operation  of  logarithmation  introduces  the  irrational 
and  imaginary  and  complex  imaginary  numbers  also,  but 
no  further  system  of  numbers. 

273.  Thus,  starting  from  the  absolute  integral  num- 
bers of  experience,  by  the  two  conditions  : 

1st.  Possibility  of  carrying  out  the  algebraic  operations 
and  their  reverse  operations  under  all  conditions, 

2d.    Permanence  of  the  laws  of  calculation, 
the  expansion  of  the  system  of  numbers  has  become  neces- 
sary, into 

Positive  and  negative  numbers, 

Integral  numbers  and  fractions, 

Rational  and  irrational  numbers, 


§  274]  COMPLEX  IMAGINARY  QUANTITIES.  405 

Real  and  imaginary  numbers  and  complex  imaginary 
numbers. 

Therewith  closes  the  Afield  of  algebra,  and  all  the  alge- 
braic operations  and  their  reverse  operations  can  be  carried 
out  irrespective  of  the  values  of  terms  entering  the  opera- 
tion. 

Thus  within  the  range  of  algebra  no  further  extension 
of  the  system  of  numbers  is  necessary  or  possible,  arid  the 
most  general  number  is 

a+jb. 

where  a  and  b  can  be   integers   or   fractions,   positive  or 
negative,  rational  or  irrational. 

ALGEBRAIC  OPERATIONS  WITH  COMPLEX  IMAGINARY 
QUANTITIES. 

274.    Definition  of  imaginary  unit: 
f2.  =  —  1. 

Complex  imaginary  number: 

A  =  a  +  jb. 
Substituting  : 

a  =  r  cos  ft 
b  =  r  sin  ft, 
it  is  A  =  r  (cos  ft  -\- j  sin  ft), 

where  r  —  V#2  -f  b\ 

a 

r  =  vector, 
ft  =  amplitude  of  complex  imaginary  number  A. 

Substituting  : 

cos  ft  - 


27 


406  APPENDIX  L  [§  275 


it  is  A  = 

I  \\n  <*>_  -I 

where   e  =  lim    1  +  -}   =  ^  -  - 

w=oo  \      r  nl  o    1  X  2  X3  X  .  .  .  Xk 

is  the  basis  of  the  natural  logarithms. 

Conjugate  numbers  : 

a  -f-  j  b  =  r  (cos  (3  -\-  j  sin  /?)  =  reJ& 

and         a  -jb  =  r(cos  [-  /?]  +7  sin  [-  £])  =  r*-./* 
it  is  («  +jb)  (a  -jb)  =  a*  +  l>*  =  r2. 

Associate  numbers  : 

a+jb  =  r(cos/8+/sin)8)  =  ?V 
and  *  +/«  =  r  (cos  [|  -  )8]  +/sin  g  -/?])  =  rtfy 

it  is  (a  +  /  b)  (b  +ja)=  j  (a2  +  £2)  =  y  ra. 

If 

it  is 


If 

it  is  a  —  0, 

6  =  0. 

275.    Addition  and  Subtraction  : 

Multiplication  : 
or     r  (cos  ft  +  j  sin  ft)  X  r'  (cos  /?'  -f-  j  sin  /3')  =  rrr  (cos 


or     r^     X  r e       =  r 

Division : 

Expansion  of  complex  imaginary  fraction,  for  rationaliza- 
tion of  denominator  or  numerator,  by  multiplication  with 
the  conjugate  quantity  : 


276,27-7]      COMPLEX  IMAGINARY  QUANTITIES.  40T 


a+jb         (*+jfy(<t  -j#)   _=  (aaf+bb')+j(baf-abr) 
a' 


_ 
(a'  +jV)  (a  -^)       (aar  +  bb')  +j  (a  br  -  b  a')  > 


r 


, 

r'(cos(3  +j  smfi')        r' 


=  r_     .(^_^ 
°r'        '  r' 


involution  : 

(a  +jb)*  =  {r  (cos  ($  +J  sin 


n  n    )  = 
evolution: 


IJa  +jb  =  ^r  (cos  p+j  sin  p)  =  ^re^ 

=  3/r{  cos^  +  /sin^  )  =  tyreJ* 
\        11  nj 

276.    Roots  of  the  Unit  : 
=l        -1 


U1 
lj 


</!=+!,       -1,     +/,      -/; 

«/T  i   L/ 

v  1  =+1?  --1?  T/»  — 

-1-/V2 


2 

n  —  =  e^ ,  k  =  0,  1,  2  ....»-  1. 


277.    Rotation  : 

In  the  complex  imaginary  plane, 
multiplication  with 


27T      .        -     -       27T 

=  cos  ---  \-j  sm  —  = 


means  rotation,  in  positive  direction,  by  1  /  n  of  a  revolution, 


408  APPENDIX  /.  [§278 

multiplication  with  (—  1)  means  reversal,  or  rotation  by  180°, 
multiplication  with  (-\-j )  means  positive  rotation  by  90°, 
multiplication  with  (— /)  means  negative  rotation  by  90°. 

278.    Complex  imaginary  plane  : 

While  the  positive  and  negative  numbers  can  be  rep- 
resented by  the  points  of  a  line,  the  complex  imaginary 
numbers  are  represented  by  the  points  of  a  plane,  with  the 
horizontal  axis  as  real  axis,  the  vertical  axis  as  imaginary 
axis.  Thus  all 

the  positive  real  numbers  are  represented  by  the  points  of  half 
axis  OA  towards  the  right ; 

the  negative  real  numbers  are  represented  by  the  points  of  half 

axis  OA'  towards  the  left ; 
the  positive  imaginary  numbers  are  represented  by  the  points  of 

half  axis  OB  upwards  ; 
the  negative  imaginary  numbers  are  represented  by  the  points  of 

half  axis  OB'  downwards  ; 
the  complex  imaginary  numbers  are  represented  by  the  points 

outside  of  the  coordinate  axes. 


APPENDIX    II. 


OSCILLATING    CURRENTS. 

INTRODUCTION. 

279.  An  electric  current  varying  periodically  between 
constant  maximum  and  minimum  values,  —  that  is,  in  equal 
time  intervals  repeating  the  same  values,  —  is  called  an 
alternating  current  if  the  arithmetic  mean  value  equals 
zero ;  and  is  called  a  pulsating  current  if  the  arithmetic 
mean  value  differs  from  zero. 

Assuming  the  wave  as  a  sine  curve,  or  replacing  it  by 
the  equivalent  sine  wave,  the  alternating  current  is  charac- 
terized .by  the  period  or  the  time  of  one  complete  cyclic 
change,  and  the  amplitude  or  the  maximum  value  of  the 
current.  Period  and  amplitude  are  constant  in  the  alter- 
nating current. 

A  very  important  class  are  the  currents  .of  constant 
period,  but  geometrically  varying  amplitude ;  that  is,  cur- 
rents in  which  the  amplitude  of  each  following  wave  bears 
to  that  of  the  preceding  wave'  a  constant  ratio.  Such 
currents  consist  of  a  series  of  waves  of  constant  length,, 
decreasing  in  amplitude,  that  is  in  strength,  in  constant 
proportion.  They  are  called  oscillating  currents  in  analogy 
with  mechanical  oscillations,  —  for  instance  of  the  pendu- 
lum,—  in  which  the  amplitude  of  the  vibration  decreases 
in  constant  proportion. 

Since  the  amplitude  of  the  oscillating  current  varies, 
constantly  decreasing,  the  oscillating  current  differs  from 

409 


410 


APPENDIX  //. 


[§  280 


the  alternating  current  in  so  far  that  it  starts  at  a  definite 
time,  and  gradually  dies  out,  reaching  zero  value  theoreti- 
cally at  infinite  time,  practically  in  a  very  short  time,  short 
even  in  comparison  with  the  time  of  one  alternating  half- 
wave.  Characteristic  constants  of  the  oscillating  current 
are  the  period  T  or  frequency  N  =  1  /  T,  the  first  ampli- 
tude and  the  ratio  of  any  two  successive  amplitudes,  the 
latter  being  called  the  decrement  of  the  wave.  The  oscil- 
lating current  will  thus  be  represented  by  the  product  of 


180 


Oscll 


900 


1389 


Fig.  181. 

•a  periodic  function,  and  a  function  decreasing  in  geometric 
proportion  with  the  time.  The  latter  is  the  exponential 
function  Af~gt. 

280.    Thus,   the  general  expression   of  the  oscillating 

current  is 

/=  ^/-K'cos  (2  vNt  -  ffl), 

since  A'-**  =  Af  A-'*  =  ie~bt. 

Where  e  =  basis  of  natural  logarithms,  the  current  may 
be  expressed 


COS 


COS      <     - 


where  <£  =  2  TT  Nt  ;  that  is,  the  period  is  represented  by  a 
complete  revolution. 


§281] 


OSCILLATING   CURRENTS. 


411 


In  the  same  way  an  oscillating  electromotive  force  will 
be  represented  by 


cos 


—  w). 


Such  an  oscillating-  electromotive  force  for  the  values 


e  =  5,     a  =  .1435  or  e-27r« 


is  represented  in  rectangular  coordinates  in  Fig.  181,  and 
in  polar  coordinates  in  Fig.  182.  As  seen  from  Fig.  181, 
the  oscillating  wave  in  rectangular  coordinates  is  tangent 
to  the  two  exponential  curves, 


y  = 


Fig.  182 


281.  In  polar  coordinates,  the  oscillating  wave  is  repre- 
sented in  Fig.  182  by  a  spiral  curve  passing  the  zero  point 
twice  per  period,  and  tangent  to  the  exponential  spiral, 


The  latter  is  called  the  envelope  of  a  system  o,.  oscillat- 
ing waves  of  which  one  is  shown  separately,  with  the  same 
constants  as  Figs.  181  and  182,  in  Fig.  183.  Its  character- 


412  APPENDIX  //.  [§  282 

istic  feature  is :    The   angle   which    any  concentric  circle 
makes  with  the  curve  y  =  et~a<b ,  is 

dv 
tan  a  =  — 4-  =  —  a, 


which  is,  therefore,  constant ;  or,  in  other  words :  "  The 
envelope  of  the  oscillating  current  is  the  exponential  spiral, 
which  is  characterized  by  a  constant  angle  of  intersection 


Fig.  183.  Fig.  184. 

with  all  concentric  circles  or  all  radii  vectores."  The  oscil- 
lating current  wave  is  the  product  of  the  sine  wave  and 
the  exponential  spiral. 

'  282.    In   Fig.   184    let  y  =  ei~a*  represent   the  expo- 
nential spiral ; 

let  z  =  e  cos  (<£  —  a) 

represent  the  sine  wave  ; 
and  let  E  =  et  ~a*  cos  (</>  —  w) 

represent  the  oscillating  wave. 
We  have  then 

dE 
tan  3  = 


—  sin  (<£  —  o>)  —  a  cos  (<j>  —  w 

cos  (<£  —  o>) 
-{tan  (<£_&)  +  «}; 


§  283]  OSCILLATING   CURREiVTS.  413 

that  is,  while  the  slope  .of  the  sine  -wave,  s  =  e  cos  (<£  —  £), 
is  represented  by 

tan  y  =  —  tan  (<£  —  w), 
the  slope  of  the  exponential  spiral  y  =  ^e~a<Ms 

tan  a  =  —  a  =  constant. 
That  of  the  oscillating  wave  E  =  ec.~a*>  cos  (<£  —  £)  is 

tan  j3  =  —  {tan  (<£  —  u>)  -f  0}. 

Hence,  it  is  increased  over  that  of  the  alternating  sine 
wave  by  the  constant  a.  The  ratio  of  the  amplitudes  of 
two  consequent  periods  is 


A  is  called  the  numerical  decrement  of  the  oscillating 
wave,  a  the  exponential  decrement  of  the  oscillating  wave, 
a  the  angular  decrement  of  the  oscillating  wave.  The 
oscillating  wave  can  be  represented  by  the  equation 

E  =  «?€-<Mana  cos   (<£  _  £). 

In  the  instance  represented  by  Figs.  181  and  182,  we 
have  A  =  .4,  a  =  .1435,  a  =  8.2°. 

•» 
Impedance  and  Admittance. 

283.  In  complex  imaginary  quantities,  the  alternating 
wave  z  =  e  cos  (*  -  5) 

is  represented  by  the  symbol 

E  =  e  (cos  w  +/  sin  w)  =  e±  +/<?2  • 

By  an  extension  of  the  meaning  of  this  symbolic  ex- 
pression, the  oscillating  wave  E  =  e€~a*  cos  (<£  —  w)  can 
be  expressed  by  the  symbol 

E  =  e  (cos  to  -f-y  sin  w)  dec  a  =  fa  +7^2)  dec  a, 
where  a  =  tan  a  is  the  exponential  decrement,  a  the  angular 
decrement,  e~2?ra  the  numerical  decrement. 


414  APPENDIX  II.  [§§  284,  285 

Inductance. 

284.    Let    r  =  resistance,    L  =  inductance,    and   x  = 
2  TT  NL  =  reactance. 

In  a  circuit  excited  by  the  oscillating  current, 


7=  z  e-«4>  Cos  (<£  —  to)  =  /  (cos  to  +/  sin  to)  dec  a  = 

(h  +  /  4)  dec  a, 
where  /\  =  z  cos  to,     /2  =  /  sin  to,     #  =  tan  a. 

We  have  then, 

The  electromotive  force  consumed  by  the  resistance  r  of 
the  circuit  J?r=r/deca. 

The  electromotive  force'  consumed  by  the  inductance  L 
of  the  circuit, 


dt  d$         d<$> 

Hence  Ex  =  —  xif.~a^  {sin  (<£  —  to)  -+-  a  cos  (<j>  —  o>)}    .   .  . 

r  /    —  a  (^ 

*^v    I-  C  •  /.  ^          •  \ 

=  --          -  sm  (</>  —  to  -f-  a). 
COS  a 

Thus,  in  symbolic  expression, 

E*=*  --  °^—{—  sin  (w  —  a)  +y  cos  (w  —  a)}  dec  a 

COS  a 

=  —  ^  /  (a  -\-  j)  (cos  w  -f  j  sin  to)  dec  a  ; 
that  is,         E^——xI(a-\-j)  dec  a  . 

Hence  the  apparent  reactance  of  the  oscillating  current 
circuit  is,  in  symbolic  expression, 

X  =  x  (a  +/)  dec  a. 

Hence   it   contains   an   energy  component   ax,  and   the 
impedance  is 

Z  =  (r  —  X)  dec  a  =  {r  —  x  (a  +_/)}  dec  a  =  (r  —  a  x  —  jx)  dec  a. 

Capacity. 

285.    Let  r  =  resistance,  C=  capacity,  and  xc  =  I/  2  TT  C 
=  capacity  reactance.     In  a  circuit  excited  by  the  oscillating 


§286]  OSCILLATING    CURRENTS.  415 

current  /,  the  electromotive  force  consumed  by  the  capacity 
C  is 


or,  by  substitution, 

Ex  =  x  \  if.~a^  cos  (<£  - 

=  - z  e~a*  {sin  (<£  —  w)  —  a  cos  (<£  —  £>' 


!+<•< 


-v»  •»  *• —  &  Q> 

sin  (<£  -  £  -  a)  ; 


(1  +  #2)  cos  a 
hence,  in  symbolic  expression, 


' — -2  (a  4~ /)  (cos  &  +/ sin  <o)  dec  a  ; 


hence, 

) /dec  a; 


that   is,  the  apparent  capacity  reactance  of  the  oscillating 
circuit  is,  in  symbolic  expression, 

C  =>  — ^ —  (—  a  -\-  f)  dec  a. 


rf^(~a+J' 

286.    We  have  then  : 

In  an  oscillating  current  circuit  of  resistance  r,  induc- 
tive reactance  x,  and  capacity  reactance  xc ,  with  an  expo- 
nential decrement  a,  the  apparent  impedance,  in  symbolic 
expression,  is  : 


=  j  r  -  a  (x  +  -^— )  -j(x  -  -^)  }  dec  a, 
C  V         1  +  a- 1         V         1  +  a- 1  ) 


416  APPENDIX  II.  [§287 

and,  absolute, 


Admittance. 
287.    Let    /==  /€_a<f>  cQs  ^  _  ^ 

Then  from  the  preceding  discussion,  the  electromotive  force 
consumed  by  resistance  r,  inductive  reactance  .r,  and  capa- 
city reactance  xc ,  is 

E  =  ie-*4>  |  cos(<£-  -       *      *c    —  sin    <£  — 


where     tan  8  — 


a 
r  —  ax  —  -         -x. 


a 
x. 


substituting  w  +  8  for  w,  and  ^  =  iza  we-  have 


=  ee-"<>  cos    <>  —  o>, 

=  —  €~a^  COS  (<^>   —  W   —   3) 


^  .  cos  8         / ,       «N    .   sin 
a<t>  }    cos  ( <^>  —  o>)  H 


hence  in  complex  quantities, 

£  =  e  (cos  o>  -J-  /  sin  <o)  dec  a, 

r        ^  (  cos  8    .     .sin  8)   , 
I  =  E\ \-j ^dec  a; 

C      Za        •  za      > 


§  288]  OSCILLA  TING   CURRENTS.  417 

or,  substituting, 


a' 

r  —  ax  --  x 


X  — 


1  +  a< 


1-M 


288.    Thus   in  complex  quantities,  for  oscillating  cur- 
rents, we  have  :  conductance, 

a 


r  —  ax— 


' 


susceptance, 


admittance,  in  absolute  values, 

~  ~ 


in  symbolic  expression, 


1  +  a2  1  + 


1  +  a*          \  1  +  a 

Since  the  impedance  is 


we  have 


418  APPENDIX  II.  [§§  289,  29O 

that  is,  the  same  relations  as  in  the  complex  quantities  in 
alternating-current  circuits,  except  that  in  the  present  case 
all  the  constants  raf  xa,  sat  g,  2,  y,  depend  upon  the  dec- 
rement a. 

Circuits  of  Zero  Impedance. 

289.  In  an  oscillating-current  circuit  of  decrement  a,  of 
resistance  r,  inductive  reactance  x,  and  capacity  reactance  xc, 
the  impedance  was  represented  in  symbolic  expression  by 

Z  =  r.  -/*„  =  (r  -aX-  ^— 
or  numerically  by 


=         r 


-  ax  -        -^  '+  (X  - 


Thus  the  inductive  reactance  x,  as  well  as  the  capacity 
reactance  xc,  do  not  represent  wattless  electromotive  forces 
as  in  an  alternating-current  circuit,  but  introduce  energy 
components  of  negative  sign 


—  a  x  — 


11      *  -  * » 
4-  a2 

that  means, 

11  In  an  oscillating-current  circuit,  the  counter  electro- 
motive force  of  self-induction  is  not  in  quadrature  behind 
the  current,  but  lags  less  than  90°,  or  a  quarter  period ;  and 
the  charging  current  of  a  condenser  is  less  than  90°,  or  a 
quarter  period,  ahead  of  the  impressed  electromotive  force." 

290.  In  consequence  of  the  existence  of  negative  en- 
ergy components  of  reactance  in  an  oscillating-current  cir- 
cuit, a  phenomenon  can  exist  which  has  no  analogy  in  an 
alternating-current  circuit ;  that  is,  under  certain  conditions 
the  total  impedance  of  the  oscillating-current  circuit  can 
equal  zero  :  =  Q 

In  this  case  we  have 


§291]  OSCILLATING    CURRENTS.  419 

substituting  in  this  equation 

x  =  l-nNL  •  xc  =  —3 : 

2 

and  expanding,  we  have* 


C  2aZ.' 

That  is, 

"  If  in  an  oscillating-current  circuit,  the  decrement 

1 


and  the  frequency  N  =  rj^traL,  the  total  impedance  of 
the  circuit  is  zero ;  that  is,  the  oscillating  current,  when 
started  once,  will  continue  without  external  energy  being 
impressed  upon  the  circuit." 

291.  The  physical  meaning  of  this  is:  "  If  upon  an 
electric  circuit  a  certain  amount  of  energy  is  impressed 
and  then  the  circuit  left  to  itself,  the  current  in  the  circuit 
will  become  oscillating,  and  the  oscillations  assume  the  fre- 
quency N  =  r/4:7raL,  and  the  decrement 

1 


a  — 


-  1 


That  is,  the  oscillating  currents  are  the  phenomena  by 
which  an  electric  circuit  of  disturbed  equilibrium  returns  to 
equilibrium. 

This  feature  shows  the  origin  of  the  oscillating  currents, 
and  the  means  to  produce  such  currents  by  disturbing 
the  equilibrium  of  the  electric  circuit  ;  for  instance,  by 
the  discharge  of  a  condenser,  by  make  and  break  of  the 
circuit,  by  sudden  electrostatic  charge,  as  lightning,  etc. 
Obviously,  the  most  important  oscillating  currents  are 


420  APPENDIX  II.  [§292 

those  flowing  in  a  circuit  of  zero  impedance,  representing 
oscillating  discharges  of  the  circuit.  Lightning  strokes 
usually  belong  to  this  class. 

Oscillating  Discharges. 

292.    The    condition    of    an    oscillating    discharge    is 
Z  =  0,  that  is,  


If  r  =  0,  that  is,  in  a  circuit  without  resistance,  we  have 
<2  =  0,  N  =  1  /  2  TT  VZ  C  ;  that  is,  the  currents  are  alter- 
nating with  no  decrement,  and  the  frequency  is  that  of 
resonance. 

If  4  H  r*  C  -  1  <  0,  that  is,  r  >  2  VZyT,  a  and  N 
become  imaginary  ;  that  is,  the  discharge  ceases  to  be  os- 
cillatory. An  electrical  discharge  assumes  an  oscillating 
nature  only,  if  r  <  2  V/,/  C.  In  the  case  r  =  2  VZ/  C  we 
have  <z  =  QO  ,  7\f  =  0  ;  that  is,  the  current  dies  out  without 
oscillation. 

From  the  foregoing  we  have  seen  that  oscillating  dis- 
charges, —  as  for  instance  the  phenomena  taking  place  if 
a  condenser  charged  to  a  given  potential  is  discharged 
through  a  given  circuit,  or  if  lightning  strikes  the  line 
circuit,  —  is  defined  by  the  equation  :  Z  —  0  dec  a. 

Since 


=  (h  +/4)  dec  a,  Er  =  Ir  dec  a, 

-_^ 


x  =  -x  I  (a  +/)  dec  a,      Exc=  -_^_  2/(-  a  +/)  dec  a, 


1  +  a 


we  have 


hence,  by  substitution, 

£xc=  x  7(—  a  -f-y)  dec  a. 


$293]  OSCILLATING   CURRENTS.  421 

The  two  constants,  il  and  z*2,  of  the  discharge,  are  deter- 
mined by  the  initial  conditions,  that  is,  the  electromotive 
force  and  the  current  at  the  time  t  =  0. 

293.  Let  a -condenser  of  capacity  C  be  discharged 
through  a  circuit  of  resistance  r  and  inductance  L.  Let 
£  =  electromotive  force  at  the  condenser  in  the  moment 
of  closing  the  circuit,  that  is,  at  the  time  /  =  0  or  <f>  =  0. 
At  this  moment  the1  current  is  zero ;  that  is, 

Z=//2,     t\  =  0. 
Since          Exc=  xl(  —  a  -f  /)  dec  a  —  e  at  </>  =  0, 


e 


we  have     x  4  Vl  +  a2  =  e  or  4  =  - 

x  Vl  +  a 
Substituting  this,  we  have, 

/> 

I  =  j  —  —  dec  a,  Er  -=je 

x  Vl  +  a? 


Ex  =  —          =-(1  —  ja)  dec  a,   Exc=—      

Vl  +  a*  Vl  +  a 

the  equations  of  the  oscillating  discharge  of  a  condenser 
of  initial  voltage  e. 

Since  x  =  2  TT  N  Z, 

a  = 


2aL 

we  have 


*  -  JL  -r-i  /4Z  _  1 

2  *  "  2  V  r2  C 

hence,  by  substitution, 


1C  1C 

1  =J'e\'J/  deca'     Er=Jer\J-j-  dec  a, 


E***^ 


422  APPENDIX  II.  [§294 


a=      , =  >  rv/_^_i 


v/ 

V^C 


the  final  equations  of  the  oscillating  discharge,  in  symbolic 
expression. 
-  • 

Oscillating  Current   Transformer. 

294.  As  an  instance  of  the  application  of  the  symbolic 
method  of  analyzing  the  phenomena  caused  by  oscillating 
currents,  the  transformation  of  such  currents  may  be  inves- 
tigated. If  an  oscillating  current  is  produced  in  a  circuit 
including  the  primary  of  a  transformer,  oscillating  currents 
will  also  flow  in  the  secondary  of  this  transformer.  In  a 
transformer  let  the  ratio  of  secondary  to  primary  turns  be/. 
Let  the  secondary  be  closed  by  a  circuit  of  total  resistance,. 
rl=  r±  -\-  7-j",  where  1\  =  external,  1\'  —  internal,  resistance, 
The  total  inductance  Ll  =  L^  -f  L^',  where  Z/  =  external,. 
LI'  —  internal,  inductance  ;  total  capacity,  Cv  Then  the 
total  admittance  of  the  secondary  circuit  is 


>.)  dec  a  = 


where  x1=  2?rArZ1=  inductive  reactance:  xcl  =  1/2  <n-NC  = 
capacity  reactance.  Let  r0  =  effecive  hysteretic  resistance,. 
LQ  =  inductance  ;  hence,  x^  =  2  -*  N  LQ  =  reactance;  hencev 

YO  =  £b  +y^o  =  7  -  -  -  —  =  admittance 
(r0  —  axo)  —jxQ 

of  the  primary  exciting  circuit  of  the  transformer  ;  that  isr 
the  admittance  of  the  primary  circuit  at  open  secondary 
.  circuit. 

As  discussed  elsewhere,  a  transformer  can  be  considered 
as  consisting  of  the  secondary  circuit  supplied  by  the  im- 
pressed electromotive  force  over  leads,  whose  impedance  is 


§294]  OSCILLATING   CURRENTS.  423 

equal  to  the  sum  .of  primary  and  secondary  transformer  im- 

pedance, and  which  are  shunted  by  the  exciting  circuit,  out- 

side of  the  secondary,  but  inside  of  the  primary  impedance. 

Let  r  =  resistance  ;  i.2L  =  inductance  ;    C  =  capacity  ; 

hence,  AT  T       .' 

x  =  2  TT  N  L  =  inductive  reactance, 

xe  =  1/2  TT  N  C  =  capacity  reactance  of  the  total  primary 
circuit,  including  the  primary  coil  of  the  transformer.  If 
EI  =  EI  dec  a  denotes  the  electromotive  force  induced  in 
the  secondary  of  the  transformer  by  the  mutual  magnetic 
flux  ;  that  is,  by  the  oscillating  magnetism  interlinked 
with  the  primary  and  secondary  coil,  we  have  7t  =  El  Yl 
dec  a  =  secondary  current. 

Hence,  //  =  /  7X  dec  a  =  p  E'  Yl  dec  a  =  primary  load 
current,  or  component  of  primary  current  corresponding  to 

secondary  current.     Also,  /0  =  —  E±   YQ   dec  a  =  primary 

P 
exciting  current  ;  hence,  the  total  primary  current  is 

1=  //  +  /0  =  ^LF0  +  />  F     dec  a. 


E  ' 
E'  =  —  -  dec  a  =  induced  primary  electromotive  force. 

/ 
Hence  the  total  primary  electromotive  force  is 

£=(£'  +  IZ)  dec  a  =  ^L  {1  +  Z  F0  +/2Z  KJ  dec  a.      * 

P 

In    an   oscillating   discharge   the   total   primary   electro- 
motive force  E  =  0  ;  that  is, 


or,  the  substitution 


r  —  a  x  --  X(.\  —  /    x 

„  o. 


424  APPENDIX  II.  [§295 

Substituting  in  this  equation,  x=%irN  '  C,  ;rc  =  l/2ir  JVC, 
etc.,  we  get  a  complex  imaginary  equation  with  the  two 
constants  a  and  N.  Separating  this  equation  in  the  real 
and  the  imaginary  parts,  we  derive  two  equations,  from 
which  the  two  constants  a  and  N  of  the  discharge  are 
calculated. 

295.  If  the  exciting  current  of  the  transformer  is  neg- 
ligible, —  that  is,  if  YQ  =  0,  the  equation  becomes  essentially 
simplified,  — 

a          \        -I  *c     \ 

—  a  x  --  xc\  —  /  \x  --  *•  — 

!+*>')         '\  1+^_Q. 

-  «*•  -  ITP*"  )  ~  j  (*>  -  r??) 

that  is, 


or,  combined,  — 

(rl-2ax1)+^(r-2ax)  =  0, 
•  =  2* 


Substituting  for  xl ,  x,  xc  l ,  xc ,  we  have 
a  = 


c  , 
1 


2  (A  irjffQ      (r1+/V)2(C1  +/2C) 
;}  dec  a, 


_-, 


P 
I  =pES  Yl  dec  a, 

7i  =  ^/  Fj  dec  a, 

the  equations  of  the   oscillating-current  transformer,  with 
£•[  as  parameter. 


INDEX. 


PAGE 

.  130 
.  123 
52,  53 
.  122 
.  57 


Advance  Angle,  of  eddy  currents 

hysteretic,  of  ironclad  circuit 

Admittance,  Chap.  vn.      .     .     . 

absolute,  of  ironclad  circuit  . 

combination  in  parallel     .     . 

combination  in  series 57 

primary,  of  transformer   ....  178 
Air-gap  in  magnetic  circuit    ....  124 
in  magnetic  circuit,  in  its  action 

on  the  hysteretic  distortion  .     .  Ill 
Algebra  of  complex  imaginary  quan- 
tities, Appendix  1 401 

Alternating  wave .     ' 11 

Alternators,      synchronizing,      Chap. 

xvii 248 

Ambiguity  of  vectors    .     .     .     .     .     .43 

Amplitude  or  maximum  value  of  wave,       7 
Analytic  method  of  treating  synchro- 
nous motor 275 

Apparent  impedance  of  transformer  .  182 
Approximations  of  line  capacity,  152,  153 

Arc,  power  factor  of 331 

pulsating,  resistance  of  ....  329 
Armature  resistance  of  induction  mo- 
tor, effect  on  starting  torque     .  226 
Asynchronous    motor,   see    induction 

motor. 
Average  value  of  wave 11 

Balanced  polyphase  system, 

Chap.  xxv.     .     .     .    356,  357,  347 

quarter-phase  system,  equations     .  396 

three-phase  system,  equations    .     .  393 

Balance  factor  of  polyphase  system     .  357 

unchanged    by   transformation   of 

polyphase  system 376 

Balance  of  polyphase  system,  in  topo- 
graphic representation     ...     46 

Capacity 5 

effect  on  higher  harmonics    .     .     .337 


PAGE 
Capacity  load,    effect  on    alternator 

field  characteristic 239 

of  line,  approximation      .     .    152,  153- 

Characteristic,  field,  of  alternator  .     .  240 

magnetic,  true  and  apparent      .     .  11& 

wave-shapes 336 

Charging  current  of  transmission  lines, 

magnitude 151 

Coefficient,  of  eddy  currents  .     .    133,  131 

of  hysteresis 116 

Combination  of   admittances  in  par- 
allel   57 

of  conductances 52 

of  impedances  in  series     ....     53 

of  resistances 52" 

of  sine  waves,  by  parallelogram  or 

polygon  on  vectors  ....   21,  22 
of  sine  waves  in  symbolic  expres- 
sion   38 

Commutator  motors,  Chap.  xix.     .     .  291 
Commutator,   sparking   at,   in   series 

and  shunt  motor 30& 

Comparison  of  systems  regarding  cop- 
per efficiency,  Chap,  xxvni      .  380 
Compensation  of  line  drop  by  shunted 

susceptance 97 

for   lagging    currents    by   shunted 

condensance 72 

Complete  resonance  in   transmission 

lines 340 

Complex  harmonic  wave 9 

Complex    imaginary   quantities,    Ap- 
pendix 1 401 

Condensance  or  capacity  reactance      .       5 
reactance   for   constant   potential, 
constant     current    transforma- 
tion  76- 

shunted,  compensation  for  lagging 

current 72 

shunted,  control  of  receiver  voltage 
in  transmission  lines    ....     96 


425 


426 


INDEX. 


Condensance,    symbolic    or    complex 

imaginary  representation     .     .     40 

Condenser    current    of    transmission 

lines,  magnitude 151 

Conductance,  Chap,  vn 52,  53 

combination  of- .     .     52 

effective 104 

of  magnetic  hysteresis      .     .     .  119 
of  mutual  inductance  ....  142 
equivalent,  of  eddy  currents  .     .     .  130 
of  receiver  circuit  supplied  over  in- 
ductive line 89 

Conductor,  eddy  currents  in  ....  138 

Constant  current  alternator   ....  246 

Constant   current,  constant  potential 

transformation 76 

Constant  potential  regulation,  of  al- 
ternator   245 

Constants,  electric,  of  magnetic  circuit 

containing  iron 125 

of  alternating-current  transformer,  190 

Control  of  receiver  potential  in  trans- 
mission lines  by  shunted  sus- 
ceptance 97 

Converter,  see  transformer. 

Converter  motor,  see  induction  motor. 

Coordinates,  rectangular  and  polar     .     19 

Copper  efficiency  of  polyphase  sys- 
tems, Chap,  xxvin 380 

Counter    E.M.F.   and   component    of 

impressed  E.M.F 24 

Creeping,  magnetic 114 

Cross  currents,  between  synchronizing 

alternators 249 

Cross  flux,  magnetic,  of  transformer, 

167,  194 

Current,  minimum,  at  given  power,  in 

synchronous  motor 282 

Cycles,  magnetic,  of  hysteresis  .     .     .  107 

Delta  connection  of  three-phase  sys- 
tem   369 

current  of  three-phase  system    .     .  371 

potential  of  three-phase  system      .  371 

Demagnetizing  effect  of  eddy  currents,  136 

Dielectric  hysteresis 144,  105 

Dielectric  hysteretic  admittance,  im- 
pedance, etc 145 

Dielectric  hysteretic  lag  angle    .     .     .  145 
Displacement  of  phase,  maximum,  in 

synchronous  motor      .     .     .     .  284 


Disruptive  phenomena 147 

Distortion    of   wave-shape   and   eddy 

currents,  Chap,  xxi 320 

Distortion  of  wave-shape  by  arc  .  .  329 
causing  decrease  of  hysteresis  .  .  343 
causing  increase  of  hysteresis  .  .  343 

in  reaction  machines 312 

of  current  wave  by  hysteresis     .     .  107 
by  quintuple  harmonic     .     .     .  335 

by  triple  harmonic 335 

Distributed  capacity,  approximation, 

152,  153 

complete  investigation      ....  155 
instance  and  particular  cases     .     .  162 
Divided  circuit,  equivalent  to  alternat- 
ing-current transformer    .     .     .  183 
Double  transformation 186 

Eddy  currents,  Chap.  xi.     .    .    129, 106 
not  affected  by  wave-shape    .     .     .  344 
Effect,  see  power,  and  output. 
Effective    resistance    and     reactance, 

Chap,  x 104 

resistance,  reactance,  conductance, 

susceptance 104 

resistance 2,  6 

value  of  wave 11,  14 

Efficiency  of  constant  potential,  con- 
stant current  transformation  by 
resistance  reactance      ....     81 
Electro-magnetic    induction,    law   of, 

Chap,  in 16 

Electro-magnetism,  as   source  of  re- 
actance   

Electrostatic  hysteresis,  see  dielectric 
hysteresis. 

induction,  or  influence 147 

Energy  components  of  current  and  of 

E.M.F 148 

component  of  reactance    ....  309 
equation  of  alternating  currents     .     23 

Equations,  of  alternator 238 

of  general  alternating-current  trans- 
former     •    .  195 

Equivalent  sine  wave    .     .     320,111,115 
Exciting  current  of  transformer      .     .  169 

Field  characteristic  of  alternator      .  240 

Flow  of  power  in  single-phase  system,  356 

in  polyphase  system     -     .     .    357,  360 

Flux  of  alternating-current  generator,  234 


INDEX. 


427 


Flux,  pulsation  of,  in  alternating-cur- 
rent generator 234 

Foucault  currents,  Chap,  xi  .     .    129,  106 
Fourier  series,  univalent  function  f>f 

time 7 

Four-phase,  see  quarter-phase. 
Four-wire   quarter-phase   system,  see 
independent  quarter-phase  sys- 
tem. 

Frequency  of  wave 7 

Frequency  setter,  with  induction  gen- 
erator     230 

Friction,  molecular 105 

Fundamental  wave,  or  first  harmonic,       8 

General  alternating-current  trans- 
former.    Chap.  xiv.   .    .   193,  195 
Generator  action  of  general  alternat- 
ing-current transformer    .     .     .  203 
Generator,  alternating-current, 

Chap,  xvi 234 

induction 229 

on  unbalanced  load,  in  topographic 

representation 48 

polar   diagram   on   non-inductive, 

inductive,  and  capacity  circuit,     28 
quarter-phase,  in  topographic  rep- 
resentation      49 

three-phase,  on  balanced  load,  in 

topographic  representation      .     46 
Graphic  method,  disadvantage  for  nu- 
merical calculation 33 

of  induction  motor 213 

of  synchronous  motor 258 

Graphic  representation,  Chap.  iv. .     .     19 

Harmonics,  even 8 

first 8 

higher,  effect  of,  Chap.  xxn.     .    .  334 
causing  resonance  in  transmis- 
sion lines 338 

effect  of  self-induction  and  ca- 
pacity     338 

of  distorted  wave Ill 

or  overtones 8 

Hedgehog  transformer 169 

Huntrng  of  parallel  running  alterna- 
tors   249 

Hysteresis,  coefficient  of 116 

decrease  due  to  distorted  wave  .     .  343 
dielectric 105,  144 


PACK 

Hysteresis,  effect  of,   on  electric  cir- 
cuit, conclusions 127 

increase  due  to  distorted  wave  .     .  343 

magnetic .'     .   105,  106 

motor 312 

Hysteretic  advance  of  phase  ....  115 

energy  current     .  , 115 

energy  current  of  transformer   .     .  170 
loss,  curve  of 114 

Imaginary  quantities,  complex,  Ap- 
pendix 1 401 

Impedance 2,  5 

apparent,  of  transformer  ....  182 
combination  in  parallel     ....    57 

combination  in  series 53 

in  series  to  circuit 68 

in  symbolic  or  complex  imaginary 

representation 39 

internal,  of  transformer    ....  178 
Increase   of    output   in   transmission 

lines,  by  shunted  susceptance  .  100 
Independent  polyphase  system  .     .     .  347 

quarter-phase  system 398 

Induced  E.M.F.,  derivation  of  formula,     16 
Inductance,  introduction  and  deriva- 
tion of  formula 18 

mutual  and  self 142 

Induction,  electrostatic,  or  influence  .  146 

generator 229 

motor  as  special  case  of  general  al- 
ternating-current   transformer, 

200,207 

motor,  Chap  xv 207 

mojtor,  graphic  method     ....  213 
motor,  numerical  instance     .     .     .  230 
motor,  symbolic  method  ....  208 
Inductive  line,  maximum  power  sup- 
plied over 87 

feeding    non-inductive,    inductive 
and   capacity   circuit,  graphic, 

23,  26,  27 
maximum  efficiency  of  transmitted 

power 93 

supplying    non-inductive    receiver 

circuit 84 

symbolic  method 40 

Inductive   load,   effect   on   alternator 

field  characteristic 239 

Influence  or  electrostatic  induction     .  146 
Instantaneous  values  of  wave,  Chap.  II.     11 


428 


INDEX. 


Insulation  strain  in  high  potential  line, 

due  to  distorted  wave-shape  .  344 
Integral  values  of  wave,  Chap.  n.  .  .  11 
Internal  impedance  of  transformer  .  178 

Intensity  of  wave 20 

Interlinked  polyphase  systems, 

Chap,  xxvi 368,  347 

quarter-phase  system 395 

Inverted  three-phase  system,  as  unbal- 
anced system 359 

flow  of  power 363 

Ironclad  inductance 124 

magnetic  circuit 106 

Iron  wire 133,  135 

J,  introduction  as  a  symbol  ....  36 

definition  as  \/ — 1 37 

Joule's  Law  of  alternating  circuits  .  6 

of  continuous  currents  ....  1 

Kirchhoff' s  Laws  of  alternating  cir- 
cuits  6,  22 

in  symbolic  or  complex  imaginary 
representation 40 

of  continuous  current  circuits   .     .       1 

Lag  angle  of  dielectric  hysteresis  .     .  145 
demagnetizing  alternator  and  mag- 
netizing synchronous  motor     .  235 
Laminated  iron    ...     .     .     .    131,135 

Lead,  magnetizing  alternator  and  de- 
magnetizing synchronous  motor  235 

Leakage 106 

current  of  transformer       ....  169 
Light  running  of  synchronous  motor,  280 
Line  capacity,  approximation    .    152,  153 
complete  investigation      ....  155 
Line,  inductive,  feeding  non-inductive, 
inductive  and  capacity  circuit, 

graphic 23,  26,  27 

load  characteristic  of    .....     95 

symbolic  method 41 

Load   characteristic   of   transmission 

line *.....    95 

Loop,  hysteretic 107 

Magnetic  hysteresis      .     .     .    105,  106 

Magnetizing  current 115 

of  transformer 170 

Magneto-motive  force  of  alternating- 
current  generator 234 

resultant  of  symmetrical  polyphase 
system 352 


Maximum  output,  or  power,  of  alter- 
nator   247 

of  induction  motor 222 

of  synchronous  motor 279 

supplied  over  inductive  line      .     .     87 

Maximum  rise  of  potential  in  receiver 

circuit  fed  by  inductive  line  .  101 
torque  of  induction  motor  .  .  .  219 
value  of  wave 11 

Mean  value  of  wave 12 

Mechanical  power  of  induction  motor, 

216,  218 

of  general  alternating-current  trans- 
former    .  * 201,  202* 

Minimum  current  at  given  power  in 

synchronous  motor     ....  282 

Molecular  friction 105 

Monocyclic  systems 363 

power  characteristic 363 

Monophase,  see  single-phase. 

Motor  action  of  general  alternating- 
current  transformer    ....  203 

commutator,  Chap,  xix 291 

repulsion 291 

series 300 

shunt 305 

synchronous,  Chap.  xvm.   .     .     .  258 

Mutual  flux  of  transformer  ....  195 

inductance 142" 

induction  of  transformers     .     .     .  167 

Non-inductive  load,  effect  on  alterna- 
tor field  characteristic     .     .     .239 
receiver  circuit  supplied  over  in- 
ductive line 84 

Non-synchronous    Motor,   see    induc- 
tion motor. 

Ohm's    Law    of    alternating-current 

circuit 2,  5 

of  continuous  currents     ....       1 
Oscillating    currents     and     E.M.Fs., 

Appendix  II 409 

Output  of  general  alternating-current 

transformer 201,  202: 

increase  of,  in  transmission  lines 

by  shunted  susceptance .     .     .  100' 
maximum  of  alternator    .     .     .  • .  247 
maximum  of  non-inductive  receiver 
circuit  supplied  over  inductive 

line 85 

maximum  of  synchronous  motor  .  279- 


INDEX. 


429 


Output    of   receiver    circuit  supplied 

over  inductive  line      ....     87 
Overtones  or  higher  harmonics  ...       8 
see  power  , 


21 


Parallelogram   of    vectors 

waves 
Parallel  operation  of  alternators, 

Chap,  xvn  ........  248 

Period  of  a  wave     ...     t     ...       7 
Phase  of  a  wave      ......     7,  20 

Phase    displacement,    maximum,    in 

synchronous  motor  .....  284 

Phase  relation  of  current  in  alternator, 

reacting  on  field     .....  235 

Polar  coordinates    .......     19 

Polar  diagram  of  transformer    .     .     .  170 
Polarization,  equivalent  to  capacity,  5,  147 
Polycyclic  systems  .     .     ..  .     .:    .     .  363 

power  characteristic    .....  363 

Polygon  of  sine  waves      .....     22 

Polyphase  systems,  general, 

Chap,  xxiii  .......  346 

interlinked,  Chap.  xxvi.      .     .     .  368 

power  characteristic    .....  363 

symmetrical,  Chap.  xxiv.     .     .     .  350 

balance  factor  of    .....  357 

transformation  of,  Chap.  xxvu.    .  376 
Power  axes  of  alternating-current  sys- 

tem     .......     .     .  366 

characteristics   of   alternating-cur- 
rent systems     ....     363,  364 

characteristics  of  synchronous  mo- 
tor ..........  278 

equation  of  alternating  circuits     .       6 
equation  of  continuous  currents    .       1 
factor  of  arc  ........  331 

factor  of  series  motor       ....  307 

flow  of,  in  single-phase  system  .     .  356 

in  polyphase  system  .     .     357,  360 

of  complex  harmonic  wave  .     .     .  341 

of  general  alternating-current  trans- 

former     ......     201,  202 

of  induction  motor      .     .     .     217,  219 
maximum,  of  alternator  ....  247 

maximum,  of  induction  motor  .     .  222 
of  repulsion  motor       .....  298 

of  synchronous  motor      ....  303 

see  output. 

Primary  admittance  of  transformer    .  178 
Pulsating  wave  ........     11 


Pulsation  of  flux  of  alternate-current 

generator 234 

Pulsation  of  magnetic  flux  distorting 

the  wave-shape 324 

of  reactance  distorting  the  wave- 
shape   328 

of  reluctance  distorting  the  wave- 
shape   327 

of  resistance  distorting  the  wave- 
shape   329 

Quarter-phase  system,  Chap.  xxx. 

395,  348,  352 

as  balanced  system 358 

copper  efficiency 380 

equations 395 

flow  of  power  .•  \  .  .  ^  .  .  361 
resultant  M.M.F.  of  .,•'...  355 
tcansformation  to  three-phase  .  .  378 
with  common  return,  unbalancing,  50 

Quintuple    harmonic,    distortion    of 

wave-shape  by 335 

Reactance,  capacity,  or  condensance,       5 
and  resistance  of  transmission  lines, 

Chap.  ix.  ...  .;.  ...  •.  .  .  83 
as  component  of  impedance  .  .  2 
capacity  in  symbolic  representation  40 
condensance  for  constant  potential, 

constant  current  transformation,  76 
effective,  Chap.  x.  .  .'.  .  .  .104 
in  series  to  circuit  .  .  .  ".  .  .61 
in  symbolic  representation  ...  38 
introduction  and  derivation  of  for- 
mula   17 

magnetic 4 

synchronous,  of  alternator    .     .     .  238 
synchronous,  of  synchronous  mo- 
tor   290 

variable  in  reaction  machines,  309,  310 
Reaction  of  lead  or  lag  in  alternator 

and  synchronous  motor  .     .     .  236 

machines,  Chap,  xx 308 

machines,  distortion  of  wave-shape,  312 

machines,  equations 315 

Reactive  component  of  pulsating  re- 
sistance     329 

Rectangular  coordinates 19 

of  vectors  in  graphic  representation 

by  polar  diagram 35 

Reduction  of  transformer  to  divided 

circuit    .  .  182 


430 


INDEX. 


Regulation  of  alternator  for  constant 

potential 245 

Reluctance  of  magnetic  circuit  of  alter- 
nator       234 

variable  in  reaction  machines,  309,  310 

Repulsion  motor 291 

equations 296 

power  of 298 

Resistance,  combination  of    ....     52 
and  reactance  of  transmission  lines, 

Chap,  ix 83 

as  component  of  impedance  ...       2 

effective,  Chap,  x 104,  2,  6 

in  series  to  circuit 58 

in  symbolic  representation    ...     38 
of  induction  motor  armature,  af- 
fecting starting  torque     .     .     .  226 
Resolving  sine  waves  in  symbolic  ex- 
pression       38 

Resonance 65 

by  higher  harmonics 115 

by  higher  harmonics  in  transmis- 
sion lines 338 

complete,  in  transmission  lines      .  340 

in  transmission  lines 101 

Resultant  magnetic  flux  of  alternator,  234 
Ring  connection  of  interlinked  poly- 
phase system 368 

of  polyphase  system 347 

Ring  current  of  interlinked  polyphase 

system 371 

Ring   potential   of    interlinked   poly- 
phase system 371 

Rise  of   potential,  maximum,  in   re- 
ceiver circuit  fed  by  inductive 

line 101 

Rotation  of  vectors 36 

Ruhmkorff  coil,  wave-shape  ....       9 

Screening  effect  of  eddy  currents  in 

iron 129 

of  eddy  current 136 

of  electric  conductor 138 

Secondary  load  on  its  action  on  hys- 

teretic  distortion Ill 

Self-inductance 3 

and  mutual  inductance      ....  142 

of  alternator 237 

Self-induction,  coefficient  of,  or  self- 
inductance      3 

effect  of  higher  harmonics     .     .     .  337 


PACK 

Self-induction,  E.M.F.  of 3. 

of  transformer 167,  194 

Series  motor 300 

equations  of 301 

phase  displacement 303 

power  ' 303 

Series  operation  of  alternators  .     .     .  250 

Sheet  iron 131,  135 

Shunted   condensance,   compensation 

for  lagging  current 72 

Shunt  motor 300,  305 

Sine  wave,  equivalent    ....    Ill,  115 

as  simplest  wave 6 

in  polar  coordinates 20 

Single-phase  system 351 

copper  efficiency 380 

flow  of  power 356,  360 

power  characteristic 363 

Slip    of    general     alternating-current 

transformer 195 

in  induction  motor 207 

Sparking  in  series  and  shunt  motors  .  306 
Star  connection  of   interlinked   poly- 
phase system 368 

of  polyphase  system 347 

Star  current  of  interlinked  polyphase 

system 371 

Star   potential    of    interlinked    poly- 
phase system 371 

Starting  resistance  of  induction  motor,  227 
torque  of  induction  motor    .     .     .  224 

Susceptance,  Chap,  vn 52,  53 

effective 104 

effective  of  mutual  inductance  .     .  143 
of   receiver   circuit   supplied   over 
inductive  line,  as  varying  out- 
put        88 

shunted,  controlling  receiver  volt- 
age in  transmission  line  ...     96 

Symbolic  method,  Chap,  v 33 

applied  to  transformer     ....  178 

representation 208 

Symmetrical  polyphase  system, 

Chap,  xxiv 350,  346 

Synchronizing  alternators, 

Chap,  xvn 248 

as  condensance  or  inductance   .     .  147 
Synchronizing  power  of   alternators, 

250,  254,  256 

Synchronism  of  induction  motor    .     .  228 
running  near,  of  induction  motor,  228 


INDEX. 


431 


Synchronous  motor,  Chap.  xvm.  .  .  258 

analytic  method 275 

discussion  of  results 288 

fundamental  equations  .  .  .  *.  276 

graphic  method  JB8 

maximum  output  .  .  * .  .  .  .  1:79 
power  characteristic 278 

Synchronous  reactance  of  alternator  .  238 
of  synchronous  motor  ....  290 

Three-phase  generator  on  balanced 

load .-    .     .     .    46 

on  unbalanced  load 48 

Three-phase  system,  Chap,  xxix., 

390,  348,  351 

as  balanced  system 358 

copper  efficiency 380 

flow  of  power     .     .     .     .     .     .     .362 

inverted,  as  unbalanced  system      .  359 

flow  of  power 352 

resultant  M.M.F 355 

topographic  representation  ...     44 
transformation  to  quarter-phase    .  378 
Three  -  phase     system,     unbalanced, 

'equations 390 

Three-wire  system,  copper  efficiency  .  381 

Time-constant 3 

Torque  of  induction  motor  ....  216 
maximum,  of  induction  motor  .     .  219 
starting,  of  induction  motor     .     .  224 
Transformation,    constant    potential, 

constant  current 76 

double.  ...  -N  •  v  -  •  •  186 
from  three-phase  to  quarter-phase 

and  inversely 378 

of  polyphase  systems,  Chap,  xxvu.,  376 

Transformer,  Chap,  xm 167 

constants  on  non-inductive  load    .  190 

equations 195 

equivalent  to  divided  circuit,  183,  184 
general  alternating-current, 

Chap,  xiv 193 

magnetic  cross-flux 194 

mo'tor,  see  induction  motor. 

numerical  instance 191 

phase  difference  in  secondary  and 

in  primary 31 

polar  diagram    ....     170,  28,  31 
stationary,  as  special  case  of  gen- 
eral   alternating-current   trans- 
former .  199 


PAGB 

Transformer,  symbolic  method      .     .  178 

Transmission  line,  load  characteristic,     95 

resistance  and  reactance,  Chap,  ix.,     83 

resonance  by  higher  harmonics      .  338 

Trigonometric  calculation  and  graphic 

representation 34 

Triple  harmonic,  distortion  of  wave- 
shape   334 

Two-phase,  see  quarter-phase. 

Unbalanced  polyphase  system, 

Chap,  xxv 356,  347 

quarter-phase  system,  equations    .  396 
three-phase  system,  equations,  390,  393 

Unbalancing  of  quarter-phase  system 

with  common  return  .     .     .     .     5"0 
of  three-phase  generator  or  system 
on  unequal  load 48 

Unequal  current  distribution  in  con- 
ductor   139 

Uniformity  of  speed  as  condition  of 

parallel  operation  of  alternators,  248 

Univalent  function  of  time  and  Fou- 
rier series 7 

Unsymmetrical  polyphase  system.     .346 

Variable  reactance  and  reluctance  in 

alternators 309,  310 

Variation  of  receiver  potential  in 
transmission  lines  by  shunted 
susceptance 96 

Vectors 19 

ambiguity  of 43 

Wattless  component  of  current  and 

of  E.M.F 148 

Wave-shape,  characteristic    ....  336 
distortion  and  its  caus.es, 

Chap,  xxi 320 

distortion  by  arc 329 

distortion  by  quintuple  harmonics,  335 
distortion  by  triple  harmonics  .     .  335 
distortion  causing  decrease  of  hys- 
teresis       343 

distortion  causing  increase  of  hys- 
teresis      ........  343 

distortion  in  reaction  machines     .  312 
Wire,  iron 133,  135 

Y  connection  of  three-phase  system  .  369 
Y  current  of  three-phase  system  .  .  371 
Y  potential  of  three-phase  system  .  .  371 


THIRD  EDITION.    .  GREA  TL  Y  ENLARGED. 

A  DICTIONARY  OF 

Electrical  Words,  Terms, 
and  Phrases. 

BY  EDWIN   J.   HOUSTON,  PH.D.   (Princeton}, 

AUTHOR  OF 

Advanced  Primers  of  Electricity  /  Electricity   One  Hundred 
Years  Ago  a/id  To-day,  etc.,  etc.,  etc. 

Cloth.    667  large  octavo  pages,  582  Illustrations. 
Price,  $5.00. 

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of  labor  involved  in  it,  may  be  formed  when  it  is  stated  that  it  contains  defini- 
tions of  about  6000  distinct  words,  terms,  or  phrases.  The  dictionary  is  not  a 
mere  word-book  ;  the  words,  terms,  and  phrases  are  invariably  followed  by  a 
short,  concise  definition,  giving  the  sense  in  which  they  are  correctly  employed, 
and  a  general  statement  of  the  principles  of  electrical  science  on  which  the  defi- 
nition is  founded.  Each  of  the  great  classes  or  divisions  of  electrical  investiga- 
tion or  utilization  comes  under  careful  and  exhaustive  treatment  ;  and  while 
close  attention  is  given  to  the  more  settled  and  hackneyed  phraseology  of  the 
older  branches  of  work,  the  newer  words  and  the  novel  departments  they  belong 
to  are  not  less  thoroughly  handled.  Every  source  of  information  has  been  re- 
ferred to,  and  while  libraries  have  been  ransacked,  the  note-book  of  the  labora- 
tory and  the  catalogue  of  the  wareroom  have  not  been  forgotten  or  neglected. 
So  far  has  the  work  been  carried  in  respect  to  the  policy  of  inclusion  that  the 
book  has  been  brought  down  to  date  by  means  of  an  appendix,  in  which  are 
placed  the  very  newest  words,  as  well  as  many  whose  rareness  of  use  had  con- 
signed them  to  obscurity  and  oblivion.  As  one  feature,  an  elaborate  system  of 
cross-references  has  been  adopted,  so  that  it  is  as  easy  to  find  the  definitions  as 
the  words,  and  atiases  are  readily  detected  and  traced.  The  typography  is  ex- 
cellent, being  large  and  bold,  and  so  arranged  that  each  word  catches  the  ej-e  at 
a  glance  by  standing  out  in  sharp  relief  from  the  page. 

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Elementary   Electro=Technical    Series. 


BY 

EDWIN  J.  HOUSTON,  PH.D., 

AND 

A.  E.  KENNELLY,  Sc.Dt 

Alternating  Electric  Currents.  Electric  Incandescent  Lighting^ 

Electric  Heating.  Electric  Motor. 

Electromagnetism.  Electric  Street  Railways. 

Electricity  in  Electro=Therapeutics.  Electric  Telephony. 

Electric  Arc  Lighting.  Electric  Telegraphy. 

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tion regarding  such  matters  in  electricity  as  cannot  be  readily  understood 
by  those  not  specially  trained  in  electro-technics.  The  general  public, 
students  of  elementary  electricity  and  the  many  interested  in  the  subject 
from  a  financial  or  other  indirect  connection,  as  well  as  electricians  desiring 
information  in  other  branches  than  their  own,  will  find  in  these  works 
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Electrical   Power  Transmission. 

-By  LofolS  BELL,  Pn.D. 

Uniform  in  size  with  "  The  Electric  Hailway  in  Theory 
and  Practice. "    Price,  $2.5O. 

The  plan  of  the  work  is  similar  to  that  of  "  The  Electric  Railway  in 
Theory  and  Practice,"  the  treatment  of  the  subject  being  non-mathematical 
and  not  involving  on  the  part  of  the  reader  a  knowledge  of  the  purely 
scientific  theories  relating  to  electrical  currents.  The  book  is  essentially 
practical  in  its  character,  and  while  primarily  an  engineering  treatise,  is 
also  intended  for  the  information  of  those  interested  in  electrical  trans- 
mission of  power,  financially  or  in  a  general  way.  The  author  has  a 
practical  acquaintance  with  the  problems  met  with  in  the  electrical  trans- 
mission of  energy  from  his  connection  with  many  of  the  most  important 
installations  yet  made  in  America,  and  in  these  pages  the  subject  is  devel- 
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actual  work.  The  first  two  chapters  review  the  fundamental  principles 
relating  to  the  generation  and  distribution  of  electrical  energy,  and  in  the 
three  succeeding  ones  their  methods  of  application  with  both  continuous 
and  alternating  currents  are  described.  The  sixth  chapter  gives  a  general 
discussion  of  the  methods  of  transformation,  the  various  considerations 
applying  to  converters  and  rotary  transformers  being  developed  and  these 
apparatus  described.  In  the  chapter  on  prime  movers  various  forms  of 
water-wheels,  gas  and  steam  engines  are  discussed  with  respect  to  their 
applicability  to  the  purpose  in  view,  and  in  the  chapter  on  hydraulic 
development  the  limitations  that  decide  the  commercial  availability1  of 
water  power  for  electrical  transmission  of  power  are  pointed  out  in  de- 
tail. The  five  succeeding  chapters  deal  with  practical  design  and  with 
construction  work — the  power-house,  line,  and  centres  of  distribution  being 
taken  up  in  turn.  The  chapter  on  the  latter  subject  will  be  found  of  par- 
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chapter  on  commercial  data  contains  the  first  information  given  as  to  costs, 
and  will,  therefore,  be  much  appreciated  by  engineers  and  others  in  decid- 
ing as  to  the  commercial  practicability  of  proposed  transmission  projects. 
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THIRD  EDITION.     EXTENSIVELY  REVISED  AND  ENLARGED. 

THE   ELECTRIC   RAILWAY 

IN  THEORY  AND  PRACTICE. 

BY  O.  T.  CROSBY  AND  DR.  LOUIS  BELL. 
Large  Octavo.    Profusely  Illustrated.    Price,  $2.5O. 


Few  technical  books  have  met  with  so  wide  an  appreciation  as  "The  Elec- 
tric Railway,"  which  has  had  the  distinction  of  being  accepted  throughout  the 
world  as  the  standard  authority  on  the  subject  of  which  it  treats.  The 
advances  in  electric  traction  made  since  the  second  edition  of  the  work  have 
been  so  notable,  that  the  authors,  in  undertaking  the  preparation  of  a  new 
edition,  found  it  necessary  to  practically  rewrite  the  book,  so  that  the  present 
edition  conforms  to  the  very  latest  knowledge  on  the  subject,  both  in  the  domain 
of  theory  and  of  practice.  The  original  purpose  Of  the  book  has',  however,  been 
strictly  adhered  to — namely,  to  place  before  those  interested  in  electric  rail- 
ways, whether  in  a  technical,  financial  or  general  way,  the  principles  upon  which 
electric  traction  is  founded  and  the  standard  methods  employed  in  their  appli- 
cation. In  view  of  the  probable  application  in  the  near  future  of  altejnating 
currents  to  electric  traction,  the  present  edition  includes  their  consideration  in 
this  relation,  thus  largely  extending  the  value  of  the  treatise.  The  recent 
developments  in  electric  locomotives  and  high-speed  electric  traction,  and  the 
application  of  electricity  to  elevated  railways  and  to  passenger  traffic  on  steam 
roads,  are  in  this  work  considered  for  the  first  time  connectedly  with  reference 
to  their  engineering  and  commercial  aspects.  In  the  first  section  of  the  book 
are  developed  the  fundamental  principles  of  electricity  upon  which  the 
apparatus  and  methods  of  operation  are  based.  The  following  section  is 
devoted  to  prime  movers,  steam,  hydraulic  and  gas — the  modern  gas  engine 
here  receiving  the  full  treatment  which  its  growing  importance  calls  for.  The 
remainder  of  the  work  is  devoted  to  the  engineering,  practical  and  commercial 
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detail.  The  plan  of  the  book,  in  fact,  includes  the  consideration  of  everything 
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from  the  engineering  and  commercial  standpoint,  and  without  the  use  of 
mathematics  or  resort  to  purely  scientific  theory. 

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PRACTICAL    CALCULATION 

*  OF 

Dynamo=Electric  Machines. 

A  MANUAL  FOR  ELECTRICAL  AND  MECHANICAL  EN- 
GINEERS, AND  A    TEXT-BOOK  FOR  STUDENTS 
OF  ELECTRO-TECHNICS. 

BY 

ALFRED  E.  WIENER,  E.E.,  M.E., 

Member  of  the  American  Institute  of  Electrical  Engineers. 

Cloth,  Illustrated.     Price,  $2.50. 

Based  upon  the  practical  data  and  tests  of  nearly  two  hundred  of  the 
best  modern  dynamos,  including  the  machines  used  at  the  recent  World's 
Fair  and  those  in  the  largest  and  most  modern  central-stations,  a  complete 
and  entirely  practical  method  of  dynamo-calculations  is  developed.  Differ- 
ing from  the  usual  text-book  methods,  in  which  the  application  of  the  vari- 
ous formulas  requires  more  or  less  experience  in  dynamo-design,  the  present 
treatise  gives  such  practical  information  in  the  form  of  original  tables  and 
formulas  derived  from  the  result  of  practical  machines  of  American  as  well 
as  European  make,  comprising  all  the  usual  types  of  field-magnets  and 
armatures,  and  ranging  through  all  commercial  sizes.  The  book  contains 
nearly  a  hundred  of  such  tables,  giving  the  values  of  the  various  constants, 
etc.,  which  enter  into  the  formulas  of  dynamo-design,  and  for  all  capacities, 
from  one-tenth  to  2000  kilowatts,  for  high  and  slow  speed,  for  bipolar  and 
multipolar  fields,  and  for  smooth  and  toothed  drum  and  ring  armatures. 
Although  intended  as  a  text-book  for  students  and  a  manual  for  practical 
dynamo-designers,  any  one  possessing  but  a  fundamental  knowledge  of 
algebra  will  be  able  to  apply  the  information  contained  in  the  book  to  the 
calculation  and  design  of  any  kind  of  a  continuous-current  dynamo. 

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THIRD  ED'ITION. 

ALTERNATING    CURRENTS. 

An  Analytical  and  Graphical  Treatment  for  Students 
and  Engineers. 

BY  FREDERICK  BEDELL,  PH.D., 

AND 

A.  C.  CREHORE,  PH.D. 
Cloth.      325  pages,    112  Illustrations.      Price,  $2.5O. 

The  present  work  is  the  first  book  that  treats  the  subject  of  alternating  cur- 
rents in  a  connected,  logical,  and  complete  manner.  The  principles  are  gradu- 
ally and  logically  developed  from  the  elementary  experiments  upon  which  they 
are  based,  and  in  a  manner  so  clear  and  simple  as  to  make  the  book  easily  read 
by  any  one  having  even  a  limited  knowledge  of  the  mathematics  involved.  By 
this  method  the  student  becomes  familiar  with  every  step  of  the  process  of 
development,  and  the  mysteries  usually  associated  with  the  theory  of  alternat- 
ing currents  are  found  to  be  rather  the  result  of  unsatistactory  treatment  than 
due  to  any  inherent  difficulty.  The  first  fourteen  chapters  contain  the  analytical 
development,  commencing  with  circuits  containing  resistance  and  self-induc- 
tion only,  resistance  and  capacity  only,  and  proceeding  to  more  complex  cir- 
cuits containing  resistance,  self-induction,  and  capacity,  and  resistance  and  dis- 
tributed capacity.  A  feature  is  the  numerical  calculations  given  as  illustrations. 
The  remaining  chapters  are  devoted  to  the  graphical  consideration  of  the  same 
subjects,  enabling  a  reader  with  little  mathematical  knowledge  to  follow  the* 
authors,  and  with  extensions  to  cases  that  are  better  treated  by  the  graphical 
than  by  the  analytical  method. 

CONTENTS. 

Part  I.  Analytical  Treatment.— Chapter  I.  Introductory  to  Circuits  Containing  Re- 
sistance and  Self-induction.— Chapter  II.  On  Harmonic  Functions.— Chap* ^r  III.  Circuits 
Containing  Resistance  and  Self-induction.— Chapter  IV.  Introductory  to  Circuits  Containing 
Resistance  and  Capacity.— Chapter  V.  Circuits  Containing  Resistance  and  Capacity.— Chap- 
ters VI,  VII,  VIII,  IX,  X,  XI.  Circuits  Containing  Resistance,  Self-induction,  and  Capacity. 
—Chapters  XII,  XIII.  Circuits  Containing  Distributed  Capacity  and  Self-induction. 

Part  II.  Graphical  Treatment.-Chapters  XIV,  XV,  XVI,  XVII.  Circuits  Containing 
Resistance  and  Self-induction.— Chapters  XVIII,  XIX.  Circuits  Containing  Resistance  and 
Capacity.— Chapter  XX.  Circuits  Containing  Resistance,  Self-induction,  and  Capacity. 

Appendix  A.  Relation  between  Practical  and  C.  G.  S.  Limits.— Appendix  B.  Some  Me- 
chanical and  Electrical  Analogies. — Appendix  C.  System  of  Notation  Adopted. 

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253  BROADWAY,  NEW  YORK, 


Electrodynamic  Machinery, 

BY 
EDWIN   J.  HOUSTON,  PH.D., 

AND 

A.  E.  KENNELLY,  Sc.D. 


Cloth.    322  Pages.    232  Illustrations.    Price,  $2.5O. 


The  advent  of  the  commercially  perfected  dynamo  has  marked  such  an 
era  in  electric  progress,  that  the  desire  naturally  exists  on  the  part  of 
students  generally,  and  of  electrical  engineers  in  particular,  to  grasp 
thoroughly  the  principles  underlying  its  constitution  and  operation. 
While  excellent  treatises  on  dynamo-electric  machinery  are  in  existence, 
yet  a  want  has  been  experienced  of  a  book  which  shall  approach  the  sub- 
ject, not  from  the  mathematical  standpoint,  but  from  an  engineering  stand- 
point, and  especially  from  such  an  engineering  standpoint  as  would  arise 
in  actual  working  with  the  apparatus.  A  book  of  this  latter  type  will  be 
found  in  the  present  work,  which  is  the  outcome  of  a  series  of  articles  on 
the  same  subject  by  these  authors  appearing  in  the  Electrical  World  during 
1894-1895.  Elettrodynamic  Machinery,  as  it  is  described  in  this  volume,  is 
limited  to  the  consideration  of  continuous-current  apparatus.  The  authors 
have  endeavored  to  explain  thoroughly  the  principles  involved  by  the  use 
of  as  simple  mathematics  as  the  case  will  permit.  In  order  to  fix  these 
principles  in  the  mind  of  the  student,  and  to  test  his  understanding  of  the 
same,  they  have  been  accompanied  by  numerous  examples  taken  from 
practical  work,  so  that  a  student  who  will  intelligently  follow  the  treat- 
ment found  in  these  pages  will  obtain  a  grasp  of  the  subject  that  it  would 
be  very  difficult  for  him  to  obtain  otherwise  without  personal  experience. 
For  this  reason  it  is  believed  that  the  book  will  serve  as  a  text-book  on 
continuous-current  dynamo-electric  machinery  for  electric-engineering 
students  of  all  grades. 

Copies  of  this  or  any  other  electrical  book  published  will  be  sent  by  mail,  POST  \£v 
PREPAID,  to  any  address  in  the  world,  on  receipt*  oj  price. 


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ALTERNATING  CURRENTS 
OF  ELECTRICITY: 

Their  Generation,  Measurement,  Distribution,  and  Application. 

BY  GISBERT  KAPP,  M.I.C.E.,  M.I.E.E. 

{Authorized  American  Edition?) 

With   an    Introduction    by    William    Stanley,   Jr. 

Cloth.    164  pages,  37  Illustrations,  and  2  plates.    $1.00. 

The  rapid  development  of  alternating  currents  and  the  great  part  they  are 
destined  to  play  in  the  transmission  of  power  have  caused  an  increased  interest 
in  the  subject,  but  unfortunately  it  has  heretofore  been  presented  in  such  a 
manner  as  to  be  beyond  the  reach  of  readers  without  a  mathematical  education. 
In  the  present  work,  the  principles  are  developed  in  a  simple  manner  that  can 
be  followed  by  any  reader,  and  the  various  applications  are  sketched  in  a  broad 
and  instructive  way  that  renders  their  understanding  an  easy  task.  The  few 
mathematical  formulas  in  the  book  are  confined  to  appendices.  Several  chap- 
ters treat  of  various  forms  of  alternating  motors,  especial  attention  being  paid 
to  the  explanation  and  discussion  of  multiphase  motors.  This  difficult  subject 
is  treated  so  lucidly  that  the  reader  is  enabled  to  form  as  clear  an  idea  of  these 
new  forms  of  motors  as  of  the  simpler  continuous  current  machines.  The 
treatment  throughout  is  thoroughly  practical,  and  the  data  and  discussion  on 
the  design  and  construction  of  apparatus  are  invaluable  to  the  electrician  and 
designer.  To  the  student  and  the  general  public  this  work  will  be  a  particular 
boon,  bringing  within  their  grasp  a  subject  of  the  greatest  importance  and 
interest. 

CONTENTS. 

Introduction  by  William  Stanley,  Jr. — Chap.  I.  Introductory. — Chap.  II.  Measurement  of 
Pressure,  Current,  and  Power.— Chap.  III.  Conditions  of  Maximum  Power.— Chap.  IV.  Alter, 
nating-current  Machines.— Chap.  V.  Mechanical  Construction  of  Alternators.— Chap.  VI. 
Description  of  Some  Alternators.— Chap.  VII  Transformers.— Chap.  VIII.  Central  Stations 
and  Distribution  of  Power.— Chap.  IX.  Examples  of.  Central  Stations.— Chap.  X.  Parallel 
Coupling  of  Alternators.— Chap.  XI.  Alternating-current  Motors.— Chap.  XII.  Self-starting 
Motors.— Chap.  XIII.  Multiphase  Currents. 

Copies  of  this  or  any  other  electrical  book  published  will  be  sent  by  mail,  POSTAGE 
PREPAID,  to  any  address  in  the  world,  on  receipt  of  price. 


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253  BROADWAY,  NEW  YORK. 


EXPERIMENTS  WITH 

Alternate    Currents 

Of  High  Potential  and  High  Frequency. 

BY  NIKOLA  TESLA. 

Cloth.    156  pages,  with  Portrait  and  35  Illustrations* 
Price,  $1.OO. 

Since  the  discovery  of  the  telephone  few  researches  in  electricity  have  created 
as  widespread  an  interest  as  those  of  Nikola  Tesla  into  alternate  currents  of 
high  potential  and  high  frequency.  Mr.  Tesla  was  accorded  the  unusual  honor 
of  an  invitation  to  repeat  his  experiments  before  distinguished  scientific  bodies 
of  London  and  Paris,  and  the  lecture  delivered  before  the  Institution  of  Elec- 
trical Engineers,  London,  is  here  presented  in  book-form.  The  field  opened 
up,  to  which  this  book  acts  as  a  guide,  is  one  in  which  the  future  may  develop 
results  of  the  most  remarkable  character,  and  perhaps  lead  to  an  entire  revision 
cf  our  present  scientific  conceptions,  with  correspondingly  broad  practical  re- 
sults. The  currents  of  enormously  high  frequency  and  voltage  generated  by 
Mr.  Tesla  developed  properties  previously  entirely  unsuspected,  and  which  pro- 
duced phenomena  of  startling  character.  No  injurious  effects  were  experi- 
enced when  the  human  body  was  subjected  to  the  highest  voltages  generated. 
Lamps  with  only  one  conductor  v/ere  rendered  incandescent,  and  others  with 
no  connection  whatever  to  the  conducting  circuit  glowed  when  merely  brought 
into  proximity  to  the  same.  The  book  in  which  Mr.  Tesla  describes  his  mar- 
vellous experiments  is  one  that  every  one  who  takes  an  interest  in  electricity  or 
its  future  should  read.  The  subject  is  popularly  treated,  and  as  the  author  is 
the  master  of  a  simple  and  agreeable  style  the  book  is  fascinating  reading.  A 
portrait  and  biographical  sketch  of  the  author  is  included. 

Copies  of  this  or  any  other  electrical  booK  published  will  be  sent  by  mail,  POSTAGE 
PREPAID,  to  any  address  in  the  world,  on  receipt  of  price. 


The  W.  J.  Johnston  Company,  Publishers, 

253  BROADWAY,  NEW  YORK. 


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The  Electrical  World.    An  Illustrated  Weekly  Review  of  Current  Progress  in 

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General  Index  to  The  Electrical  World,  1883-1896  (In  Press) 8.0O 

Atkinson,  Philip,  Ph.D.    The  Elements  of  Static  Electricity 1.5O 

Bedell,  Frederick,  Ph.D.,  and  Crehore,  Albert  C.,  Ph.D.    Alternating  Currents.  2.5O 

Bell,  Dr.  Louis.    Electrical  Transmission  of  Power 2.5O 

Cox,  Frank  P.,  B.S.    Continuous- Current  Dynamos  and  Motors 2.OO 

Crosby,  O.  T.,  and  Bell,  Dr.  Louis.   The  Electric  Railway  in  Theory  and  Practice.  2.5O 

Davis,  Charles  M.    Standard  Tables  for  Electric  Wiremen l.OO 

Foster,  H.  A.     Central-Station  Bookkeeping 2.5O 

Gerard's    Electricity.      Translated  under  the  direction   of  Dr.  Louis  Duncan. 

With  chapters  by  Messrs.  Duncan,  Steinmetz,  Kennelly  and  Hutchinson 2.5O 

tiering,  Carl.    Universal  Wiring  Computer 1  .OO 

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"  "         Electricity  at  the  Paris  Exposition  of  1889 2  OO 

"        Tables  of  Equivalents  of  Units  of  Measurement «>O 

Hopkinson,  John,  F.R.S.     Original  Papers  on  Dynamo  Machinery  and  Allied 

Subjects 1.00 

Houston,  Edwin  J.,  Ph.D.    Electrical  Measurements ,. 1  .OO 

"  "  "          Electricity  and  Magnetism l.OO 

"  "  "  Electricity  100  Years  Ago  and  To-Day l.OO 

"  "  "  Electrical  Transmission  of  Intelligence l.OO 

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Houston,  Edwin  J.,  and  Kennelly,  A.  E.    The  Electric  Motor l.OO 

"  "  "  "•     Electric  Heating l.OO 

"  "  "  "  •*        Magnetism l.OO 

"  "  "  "        The  Electric  Telephone l.OO 

•'  "  "  •*  "        Electric  Telegraphy 1.00 

••  •»  "  *»  "        Electric  Arc  Lighting l.OO 

44  "  M  ««  «        Electric  Street  Railways l.OO 


Houston,  Edwin  J.,  and  Kennelly,  A.  E.     Electrodynamic  Machinery *2.5O 

*•  **  *•  u    *        "         Electric  Incandescent  Lighting l.OO 

"  "^  "        -<  "        Alternating  Electrical  Currents l.OO 

"  "  "  *»         Electricity  in  Electro-Therapeutics..  l.OO 

Johnston's  Electrical  and  Street  Railway  Directory,    Published  annually....  5.OO 

Kapp,  Gisbert.     Alternating  Currents  of  Electricity 1  .OO 

Lightning  Flashes  and  Electric  Dashes  ^Humorous  Sketches,  Illustrated) 1.5O 

Lock  wood.  T.  D.    Practical  Information  for  Telephonists l.OO 

flartin,  T.  C.,  and  Wetzler,  Joseph.     The  Electric  Motor  and  its  Applications..  3.OO 

Haver,  William,  Jr.,  and  Davis,  Minor.    The  Quadruplex 1.5O 

rierrill,  E.  A.    Electric  Lighting  Specifications 1 .50 

"         Reference  Book  of  Tables  and  Formulae  for  Electric  Railway  Engineers.  1  .OO 
Parham  and  Shedd.    Shop  and  Road  Testing  of  Dynamos  and  Motors.  (InPress).   2.OO 

Parkhui  st ,  Lieut.  C.  D.     Dynamo  and  Motor  Building  for  Amateurs 1  .OO 

Perry ,  Nelson  W.     Electric  Railway  Motors 1  .OO 

Proceedings  of  the  National  Conference  of  Electricians 75 

Reid,  James  D.    The  Telegraph  in  America 7.OO 

Steinmetz,  C.  P.    Theory  and  Calculation  of  Alternating-Current  Phenomena.  2.5O 
Tesla,  Nikola.    Experiments  with  Alternating  Currents  of  High  Potential  and 

High  Frequency 1  .OO 

Thompson,  Prof.  Silvanus  P.    Lectures  on  the  Electromagnet l.OO 

Wiener,  Alfred  E.    Practical  Calculation  of  Dynamo-Electric  Machines 2. SO 

Wheeler,  Schuyler  S.    Chart  of  Wire  Gauges l.OO 


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